A354520 Expansion of e.g.f. exp(x) * log(cosh(x)).

0, 1, 3, 4, 0, 1, 63, 64, -1320, -1319, 49203, 49204, -2653560, -2653559, 196707423, 196707424, -19194804720, -19194804719, 2385684870723, 2385684870724, -367985503366800, -367985503366799, 68980888889771103, 68980888889771104, -15445553274667315800

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..486

Formula

a(n) = Sum_{k=1..floor(n/2)} (-1)^(k+1) * A000182(k) * binomial(n,2*k).

a(2*n) = a(2*n-1) + 1.

Prog

(PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(cosh(x)))))
(PARI) a(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
(Python)
from math import comb
from sympy import bernoulli
def A354520(n): return sum((((2<<(m:=k<<1))-2)*bernoulli(m)<<m-2)//k*comb(n, k<<1) for k in range(1, (n>>1)+1)) # Chai Wah Wu, Apr 15 2023

Crossrefs

Cf. A000182, A354518, A354519.

Sequence in context: A123685 A124917 A228550 * A189916 A025278 A200514

Adjacent sequences: A354517 A354518 A354519 * A354521 A354522 A354523

Keyword

sign

Author

Seiichi Manyama, Aug 16 2022

Status

approved