A281781 Expansion of Product_{k>=1} (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1).

1, 1, -1, 2, -1, -2, 6, -6, 3, -1, -1, 9, -18, 23, -27, 23, -1, -24, 49, -89, 121, -117, 96, -60, -18, 138, -275, 408, -525, 592, -566, 444, -181, -276, 854, -1485, 2154, -2765, 3157, -3131, 2571, -1468, -301, 2813, -5860, 9153, -12386, 15082, -16664, 16558, -14125

Offset: 0

Author

Seiichi Manyama, Apr 14 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A262811, A281683.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(4*k)/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^(4*k)*(1 - x^k)^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)

PARI

x='x+O('x^51); Vec(prod(k=1, 50, (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1))) \\ Indranil Ghosh, Apr 14 2017

Formula

G.f.: exp(Sum_{k>=1} x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, May 28 2018