In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion
,
where
is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:
![{\displaystyle E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).}](//wikimedia.org/api/rest_v1/media/math/render/svg/7d5c7e279225f2ec5e0947c7b2bf46dea201250f)
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in the OEIS) have alternating signs. Some values are:
E0 | = | 1 |
E2 | = | −1 |
E4 | = | 5 |
E6 | = | −61 |
E8 | = | 1385 |
E10 | = | −50521 |
E12 | = | 2702765 |
E14 | = | −199360981 |
E16 | = | 19391512145 |
E18 | = | −2404879675441 |
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (sequence A000364 in the OEIS). This article adheres to the convention adopted above.
In terms of Stirling numbers of the second kind
Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind[1]
[2]
![{\displaystyle E_{n}=2^{2n-1}\sum _{\ell =1}^{n}{\frac {(-1)^{\ell }S(n,\ell )}{\ell +1}}\left(3\left({\frac {1}{4}}\right)^{(\ell )}-\left({\frac {3}{4}}\right)^{(\ell )}\right),}](//wikimedia.org/api/rest_v1/media/math/render/svg/d361d1da2c8d1ad04d5287064d844cb352eca7e1)
![{\displaystyle E_{2n}=-4^{2n}\sum _{\ell =1}^{2n}(-1)^{\ell }\cdot {\frac {S(2n,\ell )}{\ell +1}}\cdot \left({\frac {3}{4}}\right)^{(\ell )},}](//wikimedia.org/api/rest_v1/media/math/render/svg/4b06fd4b10545c285218842d040ea7bb487fb695)
where
denotes the Stirling numbers of the second kind, and
denotes the rising factorial.
As a double sum
Following two formulas express the Euler numbers as double sums[3]
![{\displaystyle E_{2n}=(2n+1)\sum _{\ell =1}^{2n}(-1)^{\ell }{\frac {1}{2^{\ell }(\ell +1)}}{\binom {2n}{\ell }}\sum _{q=0}^{\ell }{\binom {\ell }{q}}(2q-\ell )^{2n},}](//wikimedia.org/api/rest_v1/media/math/render/svg/c95f62e72227b8f00a110ec08e02b823bd02fe51)
![{\displaystyle E_{2n}=\sum _{k=1}^{2n}(-1)^{k}{\frac {1}{2^{k}}}\sum _{\ell =0}^{2k}(-1)^{\ell }{\binom {2k}{\ell }}(k-\ell )^{2n}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/ff498a4b844f5a5a8b0e23a520f0dd53dc7f92dd)
As an iterated sum
An explicit formula for Euler numbers is:[4]
![{\displaystyle E_{2n}=i\sum _{k=1}^{2n+1}\sum _{\ell =0}^{k}{\binom {k}{\ell }}{\frac {(-1)^{\ell }(k-2\ell )^{2n+1}}{2^{k}i^{k}k}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/f8674d3d71a50d1fb5a268c2c178890d4b429aec)
where i denotes the imaginary unit with i2 = −1.
As a sum over partitions
The Euler number E2n can be expressed as a sum over the even partitions of 2n,[5]
![{\displaystyle E_{2n}=(2n)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq n}{\binom {K}{k_{1},\ldots ,k_{n}}}\delta _{n,\sum mk_{m}}\left(-{\frac {1}{2!}}\right)^{k_{1}}\left(-{\frac {1}{4!}}\right)^{k_{2}}\cdots \left(-{\frac {1}{(2n)!}}\right)^{k_{n}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/dbde2a76856bfe15f94d0a2bd0daa6ceed17b7c8)
as well as a sum over the odd partitions of 2n − 1,[6]
![{\displaystyle E_{2n}=(-1)^{n-1}(2n-1)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq 2n-1}{\binom {K}{k_{1},\ldots ,k_{n}}}\delta _{2n-1,\sum (2m-1)k_{m}}\left(-{\frac {1}{1!}}\right)^{k_{1}}\left({\frac {1}{3!}}\right)^{k_{2}}\cdots \left({\frac {(-1)^{n}}{(2n-1)!}}\right)^{k_{n}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/3c73d6dc96b5165940ef73c9b6a64e43d3486e8c)
where in both cases K = k1 + ··· + kn and
![{\displaystyle {\binom {K}{k_{1},\ldots ,k_{n}}}\equiv {\frac {K!}{k_{1}!\cdots k_{n}!}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b65c7383b9f4e97462957934dc731987823e6eae)
is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k1 + 4k2 + ··· + 2nkn = 2n and to k1 + 3k2 + ··· + (2n − 1)kn = 2n − 1, respectively.
As an example,
![{\displaystyle {\begin{aligned}E_{10}&=10!\left(-{\frac {1}{10!}}+{\frac {2}{2!\,8!}}+{\frac {2}{4!\,6!}}-{\frac {3}{2!^{2}\,6!}}-{\frac {3}{2!\,4!^{2}}}+{\frac {4}{2!^{3}\,4!}}-{\frac {1}{2!^{5}}}\right)\\[6pt]&=9!\left(-{\frac {1}{9!}}+{\frac {3}{1!^{2}\,7!}}+{\frac {6}{1!\,3!\,5!}}+{\frac {1}{3!^{3}}}-{\frac {5}{1!^{4}\,5!}}-{\frac {10}{1!^{3}\,3!^{2}}}+{\frac {7}{1!^{6}\,3!}}-{\frac {1}{1!^{9}}}\right)\\[6pt]&=-50\,521.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7eebe65e17bf7a388a247283db84ec59f60ed065)
As a determinant
E2n is given by the determinant
![{\displaystyle {\begin{aligned}E_{2n}&=(-1)^{n}(2n)!~{\begin{vmatrix}{\frac {1}{2!}}&1&~&~&~\\{\frac {1}{4!}}&{\frac {1}{2!}}&1&~&~\\\vdots &~&\ddots ~~&\ddots ~~&~\\{\frac {1}{(2n-2)!}}&{\frac {1}{(2n-4)!}}&~&{\frac {1}{2!}}&1\\{\frac {1}{(2n)!}}&{\frac {1}{(2n-2)!}}&\cdots &{\frac {1}{4!}}&{\frac {1}{2!}}\end{vmatrix}}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b7fdbac6c55a0760f3163eeb56bcdc953d9f8151)
As an integral
E2n is also given by the following integrals:
![{\displaystyle {\begin{aligned}(-1)^{n}E_{2n}&=\int _{0}^{\infty }{\frac {t^{2n}}{\cosh {\frac {\pi t}{2}}}}\;dt=\left({\frac {2}{\pi }}\right)^{2n+1}\int _{0}^{\infty }{\frac {x^{2n}}{\cosh x}}\;dx\\[8pt]&=\left({\frac {2}{\pi }}\right)^{2n}\int _{0}^{1}\log ^{2n}\left(\tan {\frac {\pi t}{4}}\right)\,dt=\left({\frac {2}{\pi }}\right)^{2n+1}\int _{0}^{\pi /2}\log ^{2n}\left(\tan {\frac {x}{2}}\right)\,dx\\[8pt]&={\frac {2^{2n+3}}{\pi ^{2n+2}}}\int _{0}^{\pi /2}x\log ^{2n}(\tan x)\,dx=\left({\frac {2}{\pi }}\right)^{2n+2}\int _{0}^{\pi }{\frac {x}{2}}\log ^{2n}\left(\tan {\frac {x}{2}}\right)\,dx.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1628941c1d84b5ab1f5475623da7fc2d24469b03)
The Euler numbers grow quite rapidly for large indices as
they have the following lower bound
![{\displaystyle |E_{2n}|>8{\sqrt {\frac {n}{\pi }}}\left({\frac {4n}{\pi e}}\right)^{2n}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/496e620eedd34b8c2b0882344abc01b0e2b3cc95)
The Taylor series of
is
![{\displaystyle \sum _{n=0}^{\infty }{\frac {A_{n}}{n!}}x^{n},}](//wikimedia.org/api/rest_v1/media/math/render/svg/8d8594a2cf828856a29e9a425e8e38ee65e98323)
where An is the Euler zigzag numbers, beginning with
- 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... (sequence A000111 in the OEIS)
For all even n,
![{\displaystyle A_{n}=(-1)^{\frac {n}{2}}E_{n},}](//wikimedia.org/api/rest_v1/media/math/render/svg/1b28e5b8475925656a47fce39ef6b2f80e365376)
where En is the Euler number; and for all odd n,
![{\displaystyle A_{n}=(-1)^{\frac {n-1}{2}}{\frac {2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/7ec0b17888279c88e3b90cb51470aa4ab5fb0fff)
where Bn is the Bernoulli number.
For every n,
[citation needed]
Malenfant, J. (2011). "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585 [math.NT].