A331978 E.g.f.: -log(2 - cosh(x)) (even powers only).

0, 1, 4, 46, 1114, 46246, 2933074, 263817646, 31943268634, 5009616448246, 987840438629794, 239217148602642046, 69790939492563608554, 24143849395162438623046, 9772368696995766705116914, 4575221153658910691872135246, 2453303387149157947685779986874

Offset: 0

Author

Ilya Gutkovskiy, Feb 03 2020

Keywords

• nonn

Crossrefs

Cf. A000111, A000182, A000629, A003703, A003704, A003707, A094088, A331611.

Programs

Maple

ptan := proc(n) option remember;
    if irem(n, 2) = 0 then 0 else
    add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1, 2) fi end:
A331978 := n -> ptan(2*n - 1):
seq(A331978(n), n = 0..16);  # Peter Luschny, Jun 06 2022

Mathematica

nmax = 16; Table[(CoefficientList[Series[-Log[2 - Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(0) = 0; a(n) = A094088(n) - (1/n) * Sum_{k=1..n-1} binomial(2*n,2*k) * A094088(n-k) * k * a(k).

a(n) ~ (2*n)! / (n * log(2 + sqrt(3))^(2*n)). - Vaclav Kotesovec, Feb 07 2020