A354429 - OEIS

%I #17 Nov 18 2024 17:14:41

%S 0,6,-120,3696,-168960,10830336,-929510400,103028914176,

%T -14334577213440,2446660141449216,-502760445200302080,

%U 122445316208597139456,-34878879321781771960320,11489340492300854960848896,-4333862194374775050243932160,1855989889103139616252584001536

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

%F 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))).

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

%F a(n) = (-1)^(n+1) * 6 * A059420(n). - _Andrey Zabolotskiy_, Nov 18 2024

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

%t nmax = 20; Table[(CoefficientList[Series[Tanh[x]^3, {x, 0, 2*nmax}], x]*Range[0, 2*nmax-1]!)[[k]], {k, 2, 2*nmax, 2}]

%t 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}]

%Y Cf. A000182, A059420.

%K sign

%O 0,2

%A _Vaclav Kotesovec_, May 27 2022