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

Author

Seiichi Manyama, Aug 16 2022

Keywords

• sign

Links

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

Crossrefs

Cf. A000182, A354518, A354519.

Programs

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)<>1)+1)) # Chai Wah Wu, Apr 15 2023

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.