A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

Example

log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Mathematica

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

Crossrefs

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

Sequence in context: A198038 A024255 A211045 * A211049 A321076 A166231

Adjacent sequences: A296835 A296836 A296837 * A296839 A296840 A296841

Keyword

sign

Author

Ilya Gutkovskiy, Dec 21 2017

Status

approved