A319757 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(2*k+1)/6).

1, -1, -5, -9, -6, 35, 125, 275, 291, -241, -2111, -5989, -10990, -11660, 6454, 68298, 201859, 400794, 546122, 269907, -1175825, -4890783, -11746437, -20668698, -25146121, -7959643, 63707489, 236244458, 546634684, 956731805, 1220119643, 676723572, -1964409479, -8645307595

Offset: 0

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

• sign

Crossrefs

Cf. A000330, A001157, A001158, A001159, A279215, A281156, A283263, A292386, A292387.

Programs

Maple

a:=series(mul((1-x^k)^(k*(k+1)*(2*k+1)/6),k=1..100),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 33; CoefficientList[Series[Product[(1 - x^k)^(k (2 k + 1) (k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[-Sum[x^k (1 + x^k)/(k (1 - x^k)^4), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d^2 (d + 1) (2 d + 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000330(k).

G.f.: exp(-Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^4)).

G.f.: exp(-Sum_{k>=1} (2*sigma_4(k) + 3*sigma_3(k) + sigma_2(k))*x^k/(6*k)).