A009403 Expansion of e.g.f. log(1 + tanh(x)^2), even powers only.

0, 2, -28, 992, -69088, 8110592, -1448424448, 366436769792, -124760831684608, 55014520738414592, -30501848618302701568, 20768078187214502100992, -17035983844637174375907328, 16570619538920401323784404992

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A024299.

Programs

Magma

L:=RiemannZeta();
[0] cat [-Round(4^n*(4^n-2)*(4^n-1)*Evaluate(L, 1-2*n)): n in [1..20]]; // G. C. Greubel, Jul 12 2022

Mathematica

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]^2],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 27 2013 *)

SageMath

[0]+[-4^n*(4^n-2)*(4^n-1)*zeta(1-2*n) for n in (1..20)] # G. C. Greubel, Jul 12 2022

Formula

a(n) ~ (-1)^(n+1) * 2^(4*n) * (2*n)! / (n * Pi^(2*n)). - Vaclav Kotesovec, Apr 20 2014

From G. C. Greubel, Jul 12 2022: (Start)

a(n) = 2*A024299(n).

a(n) = -4^n * (4^n - 2)*(4^n - 1)*Zeta(1-2*n), with a(0) = 0. (End)

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Aug 27 2013