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

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

Author

Seiichi Manyama, Aug 16 2022

Keywords

• nonn

Crossrefs

Cf. A000182, A354517, A354520.

Programs

PARI

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

PARI

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

Python

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

Formula

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