A318794 Constant term in the expansion of (Sum_{k=0..2*n} k*(x^k - x^(-k)))^(2*n).

1, -10, 11628, -166821980, 11017028561336, -2177623431995581080, 1017073493827776256367100, -964251586210215914665050724728, 1668594120314854076064862598821148400, -4872196290698367813554985402532435243198848

Offset: 0

Author

Seiichi Manyama, Dec 15 2018

Keywords

• sign

Crossrefs

Cf. A318793.

Programs

Mathematica

a[n_] := If[n==0, 1, Coefficient[Expand[Sum[k * (x^k - x^(-k)), {k, 0, 2n}]^(2n)], x, 0]]; Array[a, 15, 0] (* Amiram Eldar, Dec 15 2018 *)
(* Calculation of constant d: *) 64*(Sin[x]/x^2 - Cos[x]/x)^2 /. FindRoot[(2 - x^2)*Tan[x] == 2*x, {x, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Mar 17 2024 *)

PARI

{a(n) = polcoeff((sum(k=0, 2*n, k*(x^k-x^(-k))))^(2*n), 0, x)}

Formula

a(n) ~ (-1)^n * c * d^n * n^(4*n - 3/2), where d = 12.176292973966848533089025... and c = 1.04502891160415810516533... - Vaclav Kotesovec, Dec 15 2018