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
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
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]
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
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
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
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
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
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
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
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
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
References
^ 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
Possessing a specific set of other numbers
Expressible via specific sums