Euler numbers

Not to be confused with Eulerian number or Euler's number.

For other uses, see List of things named after Leonhard Euler § Numbers.

In mathematics, the Euler numbers are a sequence E_n of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

${\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}}$,

where ${\displaystyle \cosh(t)}$ is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:

${\displaystyle E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).}$

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.

Examples

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.

Explicit formulas

In terms of Stirling numbers of the second kind

The following two formulas express the Euler numbers in terms of Stirling numbers of the second kind:^[1][2]

${\displaystyle E_{n}=2^{2n-1}\sum _{\ell =1}^{n}{\frac {(-1)^{\ell }S(n,\ell )}{\ell +1}}\left(3\left({\frac {1}{4}}\right)^{\overline {\ell {\phantom {.}}}}-\left({\frac {3}{4}}\right)^{\overline {\ell {\phantom {.}}}}\right),}$

${\displaystyle E_{2n}=-4^{2n}\sum _{\ell =1}^{2n}(-1)^{\ell }\cdot {\frac {S(2n,\ell )}{\ell +1}}\cdot \left({\frac {3}{4}}\right)^{\overline {\ell {\phantom {.}}}},}$

where ${\displaystyle S(n,\ell )}$ denotes the Stirling numbers of the second kind, and ${\displaystyle x^{\overline {\ell {\phantom {.}}}}=(x)(x+1)\cdots (x+\ell -1)}$ denotes the rising factorial.

As a recursion

The Euler numbers can be defined as an recursion:

${\displaystyle E_{2n}=-\sum _{k=1}^{n}{\binom {2n}{2k}}E_{2(n-k)},}$

or alternatively:

${\displaystyle 1=-\sum _{k=1}^{n}{\binom {2n}{2k}}E_{2k},}$

Both of these recursions can be found by using the fact that.

${\displaystyle cos(x)sec(x)=1.}$

As a double sum

The following two formulas express the Euler numbers as double sums^[3]

${\displaystyle E_{2n}=(2n+1)\sum _{\ell =1}^{2n}(-1)^{\ell }{\frac {1}{2^{\ell }(\ell +1)}}{\binom {2n}{\ell }}\sum _{q=0}^{\ell }{\binom {\ell }{q}}(2q-\ell )^{2n},}$

${\displaystyle E_{2n}=\sum _{k=1}^{2n}(-1)^{k}{\frac {1}{2^{k}}}\sum _{\ell =0}^{2k}(-1)^{\ell }{\binom {2k}{\ell }}(k-\ell )^{2n}.}$

As an iterated sum

An explicit formula for Euler numbers is:^[4]

${\displaystyle E_{2n}=i\sum _{k=1}^{2n+1}\sum _{\ell =0}^{k}{\binom {k}{\ell }}{\frac {(-1)^{\ell }(k-2\ell )^{2n+1}}{2^{k}i^{k}k}},}$

where i denotes the imaginary unit with i² = −1.

As a sum over partitions

The Euler number E_2n can be expressed as a sum over the even partitions of 2n,^[5]

${\displaystyle E_{2n}=(2n)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq n}{\binom {K}{k_{1},\ldots ,k_{n}}}\delta _{n,\sum mk_{m}}\left(-{\frac {1}{2!}}\right)^{k_{1}}\left(-{\frac {1}{4!}}\right)^{k_{2}}\cdots \left(-{\frac {1}{(2n)!}}\right)^{k_{n}},}$

as well as a sum over the odd partitions of 2n − 1,^[6]

${\displaystyle E_{2n}=(-1)^{n-1}(2n-1)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq 2n-1}{\binom {K}{k_{1},\ldots ,k_{n}}}\delta _{2n-1,\sum (2m-1)k_{m}}\left(-{\frac {1}{1!}}\right)^{k_{1}}\left({\frac {1}{3!}}\right)^{k_{2}}\cdots \left({\frac {(-1)^{n}}{(2n-1)!}}\right)^{k_{n}},}$

where in both cases K = k₁ + ··· + k_n and

${\displaystyle {\binom {K}{k_{1},\ldots ,k_{n}}}\equiv {\frac {K!}{k_{1}!\cdots k_{n}!}}}$

is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k₁ + 4k₂ + ··· + 2nk_n = 2n and to k₁ + 3k₂ + ··· + (2n − 1)k_n = 2n − 1, respectively.

As an example,

${\displaystyle {\begin{aligned}E_{10}&=10!\left(-{\frac {1}{10!}}+{\frac {2}{2!\,8!}}+{\frac {2}{4!\,6!}}-{\frac {3}{2!^{2}\,6!}}-{\frac {3}{2!\,4!^{2}}}+{\frac {4}{2!^{3}\,4!}}-{\frac {1}{2!^{5}}}\right)\\[6pt]&=9!\left(-{\frac {1}{9!}}+{\frac {3}{1!^{2}\,7!}}+{\frac {6}{1!\,3!\,5!}}+{\frac {1}{3!^{3}}}-{\frac {5}{1!^{4}\,5!}}-{\frac {10}{1!^{3}\,3!^{2}}}+{\frac {7}{1!^{6}\,3!}}-{\frac {1}{1!^{9}}}\right)\\[6pt]&=-50\,521.\end{aligned}}}$

As a determinant

E_2n is given by the determinant

${\displaystyle {\begin{aligned}E_{2n}&=(-1)^{n}(2n)!~{\begin{vmatrix}{\frac {1}{2!}}&1&~&~&~\\{\frac {1}{4!}}&{\frac {1}{2!}}&1&~&~\\\vdots &~&\ddots ~~&\ddots ~~&~\\{\frac {1}{(2n-2)!}}&{\frac {1}{(2n-4)!}}&~&{\frac {1}{2!}}&1\\{\frac {1}{(2n)!}}&{\frac {1}{(2n-2)!}}&\cdots &{\frac {1}{4!}}&{\frac {1}{2!}}\end{vmatrix}}.\end{aligned}}}$

As an integral

E_2n is also given by the following integrals:

${\displaystyle {\begin{aligned}(-1)^{n}E_{2n}&=\int _{0}^{\infty }{\frac {t^{2n}}{\cosh {\frac {\pi t}{2}}}}\;dt=\left({\frac {2}{\pi }}\right)^{2n+1}\int _{0}^{\infty }{\frac {x^{2n}}{\cosh x}}\;dx\\[8pt]&=\left({\frac {2}{\pi }}\right)^{2n}\int _{0}^{1}\log ^{2n}\left(\tan {\frac {\pi t}{4}}\right)\,dt=\left({\frac {2}{\pi }}\right)^{2n+1}\int _{0}^{\pi /2}\log ^{2n}\left(\tan {\frac {x}{2}}\right)\,dx\\[8pt]&={\frac {2^{2n+3}}{\pi ^{2n+2}}}\int _{0}^{\pi /2}x\log ^{2n}(\tan x)\,dx=\left({\frac {2}{\pi }}\right)^{2n+2}\int _{0}^{\pi }{\frac {x}{2}}\log ^{2n}\left(\tan {\frac {x}{2}}\right)\,dx.\end{aligned}}}$

Congruences

W. Zhang^[7] obtained the following combinational identities concerning the Euler numbers. For any prime ${\displaystyle p}$, we have

${\displaystyle (-1)^{\frac {p-1}{2}}E_{p-1}\equiv \textstyle {\begin{cases}{\phantom {-}}0\mod p&{\text{if }}p\equiv 1{\bmod {4}};\\-2\mod p&{\text{if }}p\equiv 3{\bmod {4}}.\end{cases}}}$

W. Zhang and Z. Xu^[8] proved that, for any prime ${\displaystyle p\equiv 1{\pmod {4}}}$ and integer ${\displaystyle \alpha \geq 1}$, we have

${\displaystyle E_{\phi (p^{\alpha })/2}\not \equiv 0{\pmod {p^{\alpha }}},}$

where ${\displaystyle \phi (n)}$ is the Euler's totient function.

Lower bound

The Euler numbers grow quite rapidly for large indices, as they have the lower bound

${\displaystyle |E_{2n}|>8{\sqrt {\frac {n}{\pi }}}\left({\frac {4n}{\pi e}}\right)^{2n}.}$

Euler zigzag numbers

The Taylor series of ${\displaystyle \sec x+\tan x=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)}$ is

${\displaystyle \sum _{n=0}^{\infty }{\frac {A_{n}}{n!}}x^{n},}$

where A_n 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,

${\displaystyle A_{n}=(-1)^{\frac {n}{2}}E_{n},}$

where E_n is the Euler number, and for all odd n,

${\displaystyle A_{n}=(-1)^{\frac {n-1}{2}}{\frac {2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1}},}$

where B_n is the Bernoulli number.

For every n,

${\displaystyle {\frac {A_{n-1}}{(n-1)!}}\sin {\left({\frac {n\pi }{2}}\right)}+\sum _{m=0}^{n-1}{\frac {A_{m}}{m!(n-m-1)!}}\sin {\left({\frac {m\pi }{2}}\right)}={\frac {1}{(n-1)!}}.}$^{[citation needed]}

See also

• Bell number
• Bernoulli number
• Dirichlet beta function
• Euler–Mascheroni constant