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Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(2*k+1)/6).
0

%I #6 Apr 02 2019 05:53:34

%S 1,-1,-5,-9,-6,35,125,275,291,-241,-2111,-5989,-10990,-11660,6454,

%T 68298,201859,400794,546122,269907,-1175825,-4890783,-11746437,

%U -20668698,-25146121,-7959643,63707489,236244458,546634684,956731805,1220119643,676723572,-1964409479,-8645307595

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

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

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

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

%p 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

%t nmax = 33; CoefficientList[Series[Product[(1 - x^k)^(k (2 k + 1) (k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 33; CoefficientList[Series[Exp[-Sum[x^k (1 + x^k)/(k (1 - x^k)^4), {k, 1, nmax}]], {x, 0, nmax}], x]

%t 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}]

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

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Sep 27 2018