A354519 -id:A354519 - cp's OEIS frontend

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

Seiichi Manyama, Aug 16 2022

sign

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

Cf. A000182, A354518, A354519.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(cosh(x)))))

a(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));

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

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.

A354517 Expansion of e.g.f. cos(x)^exp(x).

Original entry on oeis.org

1, 0, -1, -3, -5, 10, 134, 742, 2325, -2820, -118756, -1138368, -7132025, -20945990, 196411214, 4438271692, 50498101545, 400644382200, 1571151012344, -16415635331328, -500300343321365, -7486919544207050, -81415563206142166, -563533196469890228

Offset: 0

Seiichi Manyama, Aug 16 2022

sign

Cf. A000248, A009189, A215515, A354518, A354519.

my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(x)^exp(x)))

a354519(n) = sum(k=1, n\2, ((-4)^k-(-16)^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, a354519(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = -Sum_{k=1..n} A354519(k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

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

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A354517 Expansion of e.g.f. cos(x)^exp(x).

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