Bernoulli number

	fraction	decimal
0	1	+1.000000000
1	±	±0.500000000
2		+0.166666666
3	0	+0.000000000
4	−	−0.033333333
5	0	+0.000000000
6		+0.023809523
7	0	+0.000000000
8	−	−0.033333333
9	0	+0.000000000
10		+0.075757575
11	0	+0.000000000
12	−	−0.253113553
13	0	+0.000000000
14		+1.166666666
15	0	+0.000000000
16	−	−7.092156862
17	0	+0.000000000
18		+54.97117794
19	0	+0.000000000
20	−	−529.1242424

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by and ; they differ only for, where and. For every odd,. For every even, is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials, with and .
The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

Notation

The superscript ± used in this article designates the two sign conventions for Bernoulli numbers. Only the term is affected:

with is the sign convention prescribed by NIST and most modern textbooks.
with is sometimes used in the older literature.

In the formulas below, one can switch from one sign convention to the other with the relation.
Since for all odd, and many formulas only involve even-index Bernoulli numbers, some authors write "" to mean. This article does not follow this notation.

History

Early history

The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.
Methods to calculate the sum of the first positive integers, the sum of the squares and of the cubes of the first positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras, Archimedes, Aryabhata, Abu Bakr al-Karaji and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham.
During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West Thomas Harriot of England, Johann Faulhaber of Germany, Pierre de Fermat and fellow French mathematician Blaise Pascal all played important roles.
Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula.
Blaise Pascal in 1654 proved Pascal's identity relating the sums of the th powers of the first positive integers for.
The Swiss mathematician Jakob Bernoulli was the first to realize the existence of a single sequence of constants which provide a uniform formula for all sums of powers.
The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the th powers for any positive integer can be seen from his comment. He wrote:
Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants.
Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834. Knuth's in-depth study of Faulhaber's formula concludes :

Reconstruction of "Summae Potestatum"

The Bernoulli numbers / were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted,, and by Bernoulli are mapped to the notation which is now prevalent as,,,. The expression means – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers. The factorial notation as a shortcut for was not introduced until 100 years later. The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter for "summa". The letter on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as. Putting things together, for positive, today a mathematician is likely to write Bernoulli's formula as:
This formula suggests setting when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6… to the modern form. Most striking in this context is the fact that the falling factorial has for the value . Thus Bernoulli's formula can be written
if, recapturing the value Bernoulli gave to the coefficient at that position.
The formula for in the first half contains an error at the last term; it should be instead of.

Definitions

Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only three of the most useful ones are mentioned:

a recursive equation,
an explicit formula,
a generating function.

For the proof of the equivalence of the three approaches see or.

Recursive definition

The Bernoulli numbers obey the sum formulas
where and denotes the Kronecker delta. Solving for gives the recursive formulas

Explicit definition

In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers, usually giving some reference in the older literature. One of them is:

Generating function

The exponential generating functions are
The generating function
is an asymptotic series. It contains the trigamma function.

Bernoulli numbers and the Riemann zeta function

The Bernoulli numbers can be expressed in terms of the Riemann zeta function:
Here the argument of the zeta function is 0 or negative.
By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained :
Now the argument of the zeta function is positive.
It then follows from and Stirling's formula that

Efficient computation of Bernoulli numbers

In some applications it is useful to be able to compute the Bernoulli numbers through modulo, where is a prime; for example to test whether Vandiver's conjecture holds for, or even just to determine whether is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least arithmetic operations would be required. Fortunately, faster methods have been developed which require only operations.
David Harvey describes an algorithm for computing Bernoulli numbers by computing modulo for many small primes, and then reconstructing via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed for. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner computed to full precision for in December 2002 and Oleksandr Pavlyk for with Mathematica in April 2008.

Applications of the Bernoulli numbers

Asymptotic analysis

Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. Assuming that is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as
This formulation assumes the convention. Using the convention the formula becomes
Here . Moreover, let denote an antiderivative of. By the fundamental theorem of calculus,
Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula
This form is for example the source for the important Euler–Maclaurin expansion of the zeta function
Here denotes the rising factorial power.
Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of the digamma function.

Sum of powers

Bernoulli numbers feature prominently in the closed form expression of the sum of the th powers of the first positive integers. For define
This expression can always be rewritten as a polynomial in of degree. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:
where denotes the binomial coefficient.
For example, taking to be 1 gives the triangular numbers .
Taking to be 2 gives the square pyramidal numbers .
Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers.
Faulhaber's formula was generalized by V. Guo and J. Zeng to a -analog.

Taylor series

The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.
;Tangent
;Cotangent
;Hyperbolic tangent
;Hyperbolic cotangent

Laurent series

The Bernoulli numbers appear in the following Laurent series :
Digamma function:

Use in topology

The Kervaire–Milnor's formula for the order of the cyclic group of diffeomorphism classes of exotic -spheres which bound parallelizable manifolds involves Bernoulli numbers. Let be the number of such exotic spheres for, then
The Hirzebruch signature theorem for the genus of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.

Connections with combinatorial numbers

The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion–exclusion principle.

Connection with Worpitzky numbers

The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function and the power function is employed. The signless Worpitzky numbers are defined as
They can also be expressed through the Stirling numbers of the second kind
A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, , , …
This representation has.
Consider the sequence,. From Worpitzky's numbers, applied to is identical to the Akiyama–Tanigawa transform applied to . This can be seen via the table:
The first row represents.
Hence for the second fractional Euler numbers / :
A second formula representing the Bernoulli numbers by the Worpitzky numbers is for
The simplified second Worpitzky's representation of the second Bernoulli numbers is:
/ = × /
which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:
The numerators of the first parentheses are .

Connection with Stirling numbers of the second kind

If denotes Stirling numbers of the second kind then one has:

where denotes the falling factorial.
If one defines the Bernoulli polynomials as :
where for are the Bernoulli numbers.
Then after the following property of binomial coefficient:
one has,
One also has following for Bernoulli polynomials,
The coefficient of in is.
Comparing the coefficient of in the two expressions of Bernoulli polynomials, one has:
which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.

Connection with Stirling numbers of the first kind

The two main formulas relating the unsigned Stirling numbers of the first kind to the Bernoulli numbers are
and the inversion of this sum
Here the number are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.
The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See /.
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes =, the autosequence is of the first kind. Example:, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: /, the second Bernoulli numbers. The Akiyama–Tanigawa transform applied to = 1/ leads to / . Hence:
See and. / are the second Euler numbers and an autosequence of the second kind.
Also valuable for / .

Connection with Pascal’s triangle

There are formulas connecting Pascal's triangle to Bernoulli numbers
where is the determinant of a n-by-n square matrix part of Pascal’s triangle whose elements are:
Example:

Connection with Eulerian numbers

There are formulas connecting Eulerian numbers to Bernoulli numbers:
Both formulae are valid for if is set to. If is set to − they are valid only for and respectively.

A binary tree representation

The Stirling polynomials are related to the Bernoulli numbers by. S. C. Woon described an algorithm to compute as a binary tree:
Woon's recursive algorithm starts by assigning to the root node. Given a node of the tree, the left child of the node is and the right child. A node is written as in the initial part of the tree represented above with ± denoting the sign of.
Given a node the factorial of is defined as
Restricted to the nodes of a fixed tree-level the sum of is, thus
For example:

Integral representation and continuation

The integral
has as special values for.
For example, and. Here, is the Riemann zeta function, and is the imaginary unit. Leonhard Euler considered these numbers and calculated

The relation to the Euler numbers and

The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the
asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers are in magnitude approximately times larger than the Bernoulli numbers. In consequence:
This asymptotic equation reveals that lies in the common root of both the Bernoulli and the Euler numbers. In fact could be computed from these rational approximations.
Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd, , it suffices to consider the case when is even.
These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to. These numbers are defined for as
and by convention. The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler in a landmark paper ‘De summis serierum reciprocarum’ and has fascinated mathematicians ever since. The first few of these numbers are
These are the coefficients in the expansion of.
The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from the sequence and scaled for use in special applications.
The expression has the value 1 if is even and 0 otherwise.
These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of when is even. The are rational approximations to and two successive terms always enclose the true value of. Beginning with the sequence starts :
These rational numbers also appear in the last paragraph of Euler's paper cited above.
Consider the Akiyama–Tanigawa transform for the sequence / :
From the second, the numerators of the first column are the denominators of Euler's formula. The first column is − ×.

An algorithmic view: the Seidel triangle

The sequence S_n has another unexpected yet important property: The denominators of S_n divide the factorial. In other words: the numbers, sometimes called Euler zigzag numbers, are integers.
Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as
These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers are given immediately by and the Bernoulli numbers are obtained from by some easy shifting, avoiding rational arithmetic.
What remains is to find a convenient way to compute the numbers. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate.

Start by putting 1 in row 0 and let denote the number of the row currently being filled
If is odd, then put the number on the left end of the row in the first position of the row, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper
At the end of the row duplicate the last number.
If is even, proceed similar in the other direction.

Seidel's algorithm is in fact much more general and was rediscovered several times thereafter.
Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers and recommended this method for computing and ‘on electronic computers using only simple operations on integers’.
V. I. Arnold rediscovered Seidel's algorithm in and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.
Triangular form:
Only, with one 1, and, with two 1s, are in the OEIS.
Distribution with a supplementary 1 and one 0 in the following rows:
This is, a signed version of. The main andiagonal is. The main diagonal is. The central column is. Row sums: 1, 1, −2, −5, 16, 61…. See. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.
The Akiyama–Tanigawa algorithm applied to / yields:
1. The first column is. Its binomial transform leads to:
The first row of this array is. The absolute values of the increasing antidiagonals are. The sum of the antidiagonals is
2. The second column is. Its binomial transform yields:
The first row of this array is. The absolute values of the second bisection are the double of the absolute values of the first bisection.
Consider the Akiyama-Tanigawa algorithm applied to / = abs) + 1 =.
The first column whose the absolute values are could be the numerator of a trigonometric function.
is an autosequence of the first kind. The corresponding array is:
The first two upper diagonals are = × . The sum of the antidiagonals is = 2 × .
− is an autosequence of the second kind, like for instance /. Hence the array:
The main diagonal, here, is the double of the first upper one, here. The sum of the antidiagonals is = 2 × . − = 2 × .

A combinatorial view: alternating permutations

Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis &. Looking at the first terms of the Taylor expansion of the trigonometric functions
and André made a startling discovery.
The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of has as coefficients the rational numbers.
André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index and the alternating permutations of even size by the Euler numbers of even index.

Related sequences

The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers:
,,,,, /. Via the second row of its inverse Akiyama–Tanigawa transform, they lead to Balmer series /.
The Akiyama–Tanigawa algorithm applied to / leads to the Bernoulli numbers /, /, or without, named intrinsic Bernoulli numbers.
Hence another link between the intrinsic Bernoulli numbers and the Balmer series via .
= 0, 2, 1, 6,… is a permutation of the non-negative numbers.
The terms of the first row are f =. 2, f is an autosequence of the second kind. 3/2, f leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5... = 1/2 + log 2.
Consider g = 1/2 - 1 / = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:
0, g, is an autosequence of the second kind.
Euler / without the second term are the fractional intrinsic Euler numbers The corresponding Akiyama transform is:
The first line is. preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are preceded by 0. The difference table is:

Arithmetical properties of the Bernoulli numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as for integers provided for the expression is understood as the limiting value and the convention is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that is a prime number if and only if is congruent to −1 modulo. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Kummer theorems

The Bernoulli numbers are related to Fermat's Last Theorem by Kummer's theorem, which says:
Prime numbers with this property are called regular primes. Another classical result of Kummer are the following congruences.
A generalization of these congruences goes by the name of -adic continuity.

-adic continuity

If, and are positive integers such that and are not divisible by and, then
Since, this can also be written
where and, so that and are nonpositive and not congruent to 1 modulo. This tells us that the Riemann zeta function, with taken out of the Euler product formula, is continuous in the -adic numbers on odd negative integers congruent modulo to a particular, and so can be extended to a continuous function for all -adic integers, the -adic zeta function.

Ramanujan's congruences

The following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition:

Von Staudt–Clausen theorem

The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt and Thomas Clausen independently in 1840. The theorem states that for every,
is an integer. The sum extends over all primes for which divides.
A consequence of this is that the denominator of is given by the product of all primes for which divides. In particular, these denominators are square-free and divisible by 6.

Why do the odd Bernoulli numbers vanish?

The sum
can be evaluated for negative values of the index. Doing so will show that it is an odd function for even values of, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that is 0 for even and ; and that the term for is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question.
From the von Staudt–Clausen theorem it is known that for odd the number is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets
as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let be the number of surjective maps from to, then. The last equation can only hold if
This equation can be proved by induction. The first two examples of this equation are
Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.

A restatement of the Riemann hypothesis

The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis which uses only the Bernoulli number. In fact Marcel Riesz proved that the RH is equivalent to the following assertion:
Here is the Riesz function
denotes the rising factorial power in the notation of D. E. Knuth. The numbers occur frequently in the study of the zeta function and are significant because is a -integer for primes where does not divide. The are called divided Bernoulli numbers.

Generalized Bernoulli numbers

The generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, that are related to special values of Dirichlet -functions in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.
Let be a Dirichlet character modulo. The generalized Bernoulli numbers attached to are defined by
Apart from the exceptional, we have, for any Dirichlet character, that if.
Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers :
where is the Dirichlet -function of .

Appendix

Assorted identities

Popular movies

The Hunger Games (film) - 2012 American dystopian action thriller science fiction-adventure film directed by Gary Ross and based on Suzanne Collins’s 2008 novel of the same name. It is the first insta...
untitled Captain Marvel sequel - part of Marvel Cinematic Universe....
Killers of the Flower Moon (film project) - Killers of the Flower Moon - film project in United States of America. It was presented as drama, detective fiction, thriller. The film project starred Leonardo Dicaprio, Robert De Niro. Director of...
Five Nights at Freddy's (film) - Five Nights at Freddy's - film published in 2017 in United States of America. Scenarist of the film - Scott Cawthon....

Popular books

Book of Revelation - The Book of Revelation is the final book of the New Testament, and consequently is also the final book of the Christian Bible. Its title is derived from the first word of the Koine Greek text: apok...
Book of Genesis - account of the creation of the world, the early history of humanity, Israel's ancestors and the origins...
Gospel of Matthew - The Gospel According to Matthew is the first book of the New Testament and one of the three synoptic gospels. It tells how Israel's Messiah, rejected and executed in Israel, pronounces judgement on ...
Michelin Guide - Michelin Guides are a series of guide books published by the French tyre company Michelin for more than a century. The term normally refers to the annually published Michelin Red Guide , the oldest...
Psalms - The Book of Psalms , commonly referred to simply as Psalms , the Psalter or "the Psalms", is the first book of the Ketuvim , the third section of the Hebrew Bible, and thus a book of th...
Ecclesiastes - Ecclesiastes is one of 24 books of the Tanakh , where it is classified as one of the Ketuvim . Originally written c. 450–200 BCE, it is also among the canonical Wisdom literature of the Old Tes...
The 48 Laws of Power - non-fiction book by American author Robert Greene. The book...

Popular television series

The Crown (TV series) - historical drama web television series about the reign of Queen Elizabeth II, created and principally written by Peter Morgan, and produced by Left Bank Pictures and Sony Pictures Tel...
Friends - American sitcom television series, created by David Crane and Marta Kauffman, which aired on NBC from September 22, 1994, to May 6, 2004, lasting ten seasons. With an ensemble cast sta...
Young Sheldon - spin-off prequel to The Big Bang Theory and begins with the character Sheldon...
Modern Family - American television mockumentary family sitcom created by Christopher Lloyd and Steven Levitan for the American Broadcasting Company. It ran for eleven seasons, from September 23...
Loki (TV series) - upcoming American web television miniseries created for Disney+ by Michael Waldron, based on the Marvel Comics character of the same name. It is set in the Marvel Cinematic Universe, shar...
Game of Thrones - American fantasy drama television series created by David Benioff and D. B. Weiss for HBO. It...
Shameless (American TV series) - American comedy-drama television series developed by John Wells which debuted on Showtime on January 9, 2011. It...