A354429 Expansion of e.g.f. tanh(x)^3 (odd powers only).

0, 6, -120, 3696, -168960, 10830336, -929510400, 103028914176, -14334577213440, 2446660141449216, -502760445200302080, 122445316208597139456, -34878879321781771960320, 11489340492300854960848896, -4333862194374775050243932160, 1855989889103139616252584001536

Offset: 0

Author

Vaclav Kotesovec, May 27 2022

Keywords

• sign

Examples

			tanh(x)^3 = x^3 - x^5 + 11*x^7/15 - 88*x^9/189 + ...
		

Crossrefs

Cf. A000182, A059420.

Programs

Mathematica

nmax = 20; Table[(CoefficientList[Series[Tanh[x]^3, {x, 0, 2*nmax}], x]*Range[0, 2*nmax-1]!)[[k]], {k, 2, 2*nmax, 2}]
nmax = 20; Table[Sum[-k * 2^(2*n + 3) * (2^(k + 1) - 1) * (2^(2*n + 3 - k) - 1) * BernoulliB[k + 1] * BernoulliB[2*n + 3 - k] * Binomial[2*n + 3, k] / ((k + 1) * (n + 1) * (2*n + 3)), {k, 2, 2*n + 1}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=2..2*n+1} (-k * 2^(2*n+3) * (2^(k+1) - 1) * (2^(2*n + 3 - k) - 1) * Bernoulli(k+1) * Bernoulli(2*n + 3 - k) * binomial(2*n + 3, k) / ((k+1) * (n+1) * (2*n+3))).

a(n) ~ (-1)^(n+1) * 2^(4*n + 8) * n^(2*n + 7/2) / (exp(2*n) * Pi^(2*n + 7/2)).

a(n) = (-1)^(n+1) * 6 * A059420(n). - Andrey Zabolotskiy, Nov 18 2024