Integers occurring in the coefficients of the Taylor series of 1/cosh t
In mathematics , the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS ) defined by the Taylor series expansion
1 cosh t = 2 e t + e − t = ∑ n = 0 ∞ E n n ! ⋅ t n {\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}} , where cosh ( t ) {\displaystyle \cosh(t)} is the hyperbolic cosine function . The Euler numbers are related to a special value of the Euler polynomials , namely:
E n = 2 n E n ( 1 2 ) . {\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 [ edit ] The odd-indexed Euler numbers are all zero . The even-indexed ones (sequence A028296 in the OEIS ) have alternating signs. Some values are:
E 0 = 1 E 2 = −1 E 4 = 5 E 6 = −61 E 8 = 1385 E 10 = −50521 E 12 = 2702 765 E 14 = −199360 981 E 16 = 19391 512 145 E 18 = −2404 879 675 441
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 [ edit ] In terms of Stirling numbers of the second kind [ edit ] Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind [1] [2]
E n = 2 2 n − 1 ∑ ℓ = 1 n ( − 1 ) ℓ S ( n , ℓ ) ℓ + 1 ( 3 ( 1 4 ) ( ℓ ) − ( 3 4 ) ( ℓ ) ) , {\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),} E 2 n = − 4 2 n ∑ ℓ = 1 2 n ( − 1 ) ℓ ⋅ S ( 2 n , ℓ ) ℓ + 1 ⋅ ( 3 4 ) ( ℓ ) , {\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 )},} where S ( n , ℓ ) {\displaystyle S(n,\ell )} denotes the Stirling numbers of the second kind , and x ( ℓ ) = ( x ) ( x + 1 ) ⋯ ( x + ℓ − 1 ) {\displaystyle x^{(\ell )}=(x)(x+1)\cdots (x+\ell -1)} denotes the rising factorial .
As a double sum [ edit ] Following two formulas express the Euler numbers as double sums[3]
E 2 n = ( 2 n + 1 ) ∑ ℓ = 1 2 n ( − 1 ) ℓ 1 2 ℓ ( ℓ + 1 ) ( 2 n ℓ ) ∑ q = 0 ℓ ( ℓ q ) ( 2 q − ℓ ) 2 n , {\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},} E 2 n = ∑ k = 1 2 n ( − 1 ) k 1 2 k ∑ ℓ = 0 2 k ( − 1 ) ℓ ( 2 k ℓ ) ( k − ℓ ) 2 n . {\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 [ edit ] An explicit formula for Euler numbers is:[4]
E 2 n = i ∑ k = 1 2 n + 1 ∑ ℓ = 0 k ( k ℓ ) ( − 1 ) ℓ ( k − 2 ℓ ) 2 n + 1 2 k i k k , {\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 i 2 = −1 .
As a sum over partitions [ edit ] The Euler number E 2n can be expressed as a sum over the even partitions of 2n ,[5]
E 2 n = ( 2 n ) ! ∑ 0 ≤ k 1 , … , k n ≤ n ( K k 1 , … , k n ) δ n , ∑ m k m ( − 1 2 ! ) k 1 ( − 1 4 ! ) k 2 ⋯ ( − 1 ( 2 n ) ! ) k n , {\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]
E 2 n = ( − 1 ) n − 1 ( 2 n − 1 ) ! ∑ 0 ≤ k 1 , … , k n ≤ 2 n − 1 ( K k 1 , … , k n ) δ 2 n − 1 , ∑ ( 2 m − 1 ) k m ( − 1 1 ! ) k 1 ( 1 3 ! ) k 2 ⋯ ( ( − 1 ) n ( 2 n − 1 ) ! ) k n , {\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 = k 1 + ··· + kn and
( K k 1 , … , k n ) ≡ K ! k 1 ! ⋯ k n ! {\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 k s to 2k 1 + 4k 2 + ··· + 2nkn = 2n and to k 1 + 3k 2 + ··· + (2n − 1)kn = 2n − 1 , respectively.
As an example,
E 10 = 10 ! ( − 1 10 ! + 2 2 ! 8 ! + 2 4 ! 6 ! − 3 2 ! 2 6 ! − 3 2 ! 4 ! 2 + 4 2 ! 3 4 ! − 1 2 ! 5 ) = 9 ! ( − 1 9 ! + 3 1 ! 2 7 ! + 6 1 ! 3 ! 5 ! + 1 3 ! 3 − 5 1 ! 4 5 ! − 10 1 ! 3 3 ! 2 + 7 1 ! 6 3 ! − 1 1 ! 9 ) = − 50 521. {\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 [ edit ] E 2n is given by the determinant
E 2 n = ( − 1 ) n ( 2 n ) ! | 1 2 ! 1 1 4 ! 1 2 ! 1 ⋮ ⋱ ⋱ 1 ( 2 n − 2 ) ! 1 ( 2 n − 4 ) ! 1 2 ! 1 1 ( 2 n ) ! 1 ( 2 n − 2 ) ! ⋯ 1 4 ! 1 2 ! | . {\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 [ edit ] E 2n is also given by the following integrals:
( − 1 ) n E 2 n = ∫ 0 ∞ t 2 n cosh π t 2 d t = ( 2 π ) 2 n + 1 ∫ 0 ∞ x 2 n cosh x d x = ( 2 π ) 2 n ∫ 0 1 log 2 n ( tan π t 4 ) d t = ( 2 π ) 2 n + 1 ∫ 0 π / 2 log 2 n ( tan x 2 ) d x = 2 2 n + 3 π 2 n + 2 ∫ 0 π / 2 x log 2 n ( tan x ) d x = ( 2 π ) 2 n + 2 ∫ 0 π x 2 log 2 n ( tan x 2 ) d x . {\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 [ edit ] W. Zhang[7] obtained the following combinational identities concerning the Euler numbers, for any prime p {\displaystyle p} , we have
( − 1 ) p − 1 2 E p − 1 ≡ { 0 mod p if p ≡ 1 mod 4 ; − 2 mod p if p ≡ 3 mod 4 . {\displaystyle (-1)^{\frac {p-1}{2}}E_{p-1}\equiv \textstyle {\begin{cases}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 p ≡ 1 ( mod 4 ) {\displaystyle p\equiv 1{\pmod {4}}} and integer α ≥ 1 {\displaystyle \alpha \geq 1} , we have
E ϕ ( p α ) / 2 ≢ 0 ( mod p α ) {\displaystyle E_{\phi (p^{\alpha })/2}\not \equiv 0{\pmod {p^{\alpha }}}} where ϕ ( n ) {\displaystyle \phi (n)} is the Euler's totient function .
Asymptotic approximation [ edit ] The Euler numbers grow quite rapidly for large indices as they have the following lower bound
| E 2 n | > 8 n π ( 4 n π e ) 2 n . {\displaystyle |E_{2n}|>8{\sqrt {\frac {n}{\pi }}}\left({\frac {4n}{\pi e}}\right)^{2n}.} Euler zigzag numbers [ edit ] The Taylor series of sec x + tan x = tan ( π 4 + x 2 ) {\displaystyle \sec x+\tan x=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)} is
∑ n = 0 ∞ A n n ! x n , {\displaystyle \sum _{n=0}^{\infty }{\frac {A_{n}}{n!}}x^{n},} 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 ,
A n = ( − 1 ) n 2 E n , {\displaystyle A_{n}=(-1)^{\frac {n}{2}}E_{n},} where En is the Euler number; and for all odd n ,
A n = ( − 1 ) n − 1 2 2 n + 1 ( 2 n + 1 − 1 ) B n + 1 n + 1 , {\displaystyle A_{n}=(-1)^{\frac {n-1}{2}}{\frac {2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1}},} where Bn is the Bernoulli number .
For every n ,
A n − 1 ( n − 1 ) ! sin ( n π 2 ) + ∑ m = 0 n − 1 A m m ! ( n − m − 1 ) ! sin ( m π 2 ) = 1 ( n − 1 ) ! . {\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 [ edit ] References [ edit ] ^ Jha, Sumit Kumar (2019). "A new explicit formula for Bernoulli numbers involving the Euler number" . Moscow Journal of Combinatorics and Number Theory . 8 (4): 385–387. doi :10.2140/moscow.2019.8.389 . S2CID 209973489 . ^ Jha, Sumit Kumar (15 November 2019). "A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind" . ^ Wei, Chun-Fu; Qi, Feng (2015). "Several closed expressions for the Euler numbers" . Journal of Inequalities and Applications . 219 (2015). doi :10.1186/s13660-015-0738-9 . ^ Tang, Ross (2012-05-11). "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" (PDF) . Archived (PDF) from the original on 2014-04-09. ^ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers" . Integers . 8 (1): A1. ^ Malenfant, J. (2011). "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv :1103.1585 [math.NT ]. ^ Zhang, W.P. (1998). "Some identities involving the Euler and the central factorial numbers" (PDF) . Fibonacci Quarterly . 36 (4): 154–157. Archived (PDF) from the original on 2019-11-23. ^ Zhang, W.P.; Xu, Z.F. (2007). "On a conjecture of the Euler numbers" . Journal of Number Theory . 127 (2): 283–291. doi :10.1016/j.jnt.2007.04.004 . External links [ edit ]
Possessing a specific set of other numbers
Expressible via specific sums