A354517 -id:A354517 - cp's OEIS frontend

A354518 Expansion of e.g.f. cosh(x)^exp(x).

1, 0, 1, 3, 7, 30, 166, 798, 4117, 27660, 196756, 1328448, 9866407, 86205210, 759842266, 6460661028, 60841732777, 651349676280, 6795873687496, 67981177154688, 770224145659627, 9854500496860470, 116983085896035646, 1301594922821009028, 17440543467561038557

Offset: 0

Seiichi Manyama, Aug 16 2022

sign

a(39) is negative. - Vaclav Kotesovec, Aug 17 2022

Seiichi Manyama, Table of n, a(n) for n = 0..486

Cf. A000248, A003727, A215518, A354517, A354520.

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

a354520(n) = sum(k=1, n\2, (16^k-4^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, a354520(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A354520(k) * binomial(n-1,k-1) * a(n-k).

A354519 Expansion of e.g.f. exp(x) * log(sec(x)).

Original entry on oeis.org

0, 1, 3, 8, 20, 61, 203, 888, 4080, 24001, 140283, 1028048, 7248020, 63374221, 522164243, 5299033488, 49924707840, 576514338721, 6110861416083, 79100066353208, 931434877343540, 13355627237749501, 172948115797623803, 2720827878727067208, 38424408320191299120

Offset: 1

Seiichi Manyama, Aug 16 2022

nonn

Cf. A000182, A354517, A354520.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(1/cos(x)))))

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

from math import comb
from sympy import bernoulli
def A354519(n): return sum(abs(((2-(2<<(m:=k<<1)))*bernoulli(m)<>1)+1)) # Chai Wah Wu, Apr 15 2023

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

a(n) ~ 2^(n + 1/2) * (exp(Pi/2) + (-1)^n/exp(Pi/2)) * n^(n - 1/2) / (Pi^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Aug 17 2022

cp's OEIS Frontend

A354518 Expansion of e.g.f. cosh(x)^exp(x).

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A354519 Expansion of e.g.f. exp(x) * log(sec(x)).

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