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Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).
2

%I #5 May 09 2021 11:17:00

%S 1,1,5,20,60,190,561,1651,4720,13300,36716,99872,267836,708890,

%T 1854255,4796273,12279445,31135188,78236006,194921680,481758832,

%U 1181675902,2877646681,6959866116,16723591530,39934902812,94795718409,223741936855,525206126933,1226393510220

%N Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).

%F G.f.: exp( Sum_{k>=1} (-1)^(k+1) * x^k / (k*(1 - x^k)^5) ).

%t nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 4], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]

%Y Cf. A000332, A000391, A028377, A258343, A305206, A344100, A344101.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 09 2021