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A343200 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,3).

%I A343200 #27 May 12 2021 09:15:21
%S A343200 1,4,16,64,221,736,2338,7132,21093,60652,170172,467140,1257571,
%T A343200 3325824,8654576,22189340,56116043,140122760,345769094,843827436,
%U A343200 2038017983,4874329024,11550814704,27134195608,63215468883,146120097736,335227455982,763592477104,1727482413548
%N A343200 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,3).
%H A343200 Vaclav Kotesovec, <a href="/A343200/b343200.txt">Table of n, a(n) for n = 0..7650</a>
%F A343200 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A033488(d) ) * a(n-k).
%F A343200 a(n) ~ (3*zeta(5))^(1/10) / (2^(7/10) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-469*log(2)/720 - 2401*Pi^16 / (656100000000*zeta(5)^3) + 539*Pi^8*zeta(3) / (8100000*zeta(5)^2) - 7*Pi^6 / (27000*zeta(5)) - 121*zeta(3)^2 / (600*zeta(5)) + (343*Pi^12 / (303750000 * 2^(3/5) * 15^(1/5) * zeta(5)^(11/5)) - 77*Pi^4*zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * zeta(5)^(6/5)) + Pi^2 / (6*2^(3/5) * (15*zeta(5))^(1/5))) * n^(1/5) + (-49*Pi^8 / (270000 * 2^(1/5) * 15^(2/5) * zeta(5)^(7/5)) + 11*zeta(3) / (4*2^(1/5) * (15*zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (90*2^(4/5) * (15*zeta(5))^(3/5))) * n^(3/5) + (5*(15*zeta(5))^(1/5) / (4*2^(2/5))) * n^(4/5)). - _Vaclav Kotesovec_, May 12 2021
%t A343200 nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 3], {k, 1, nmax}], {x, 0, nmax}], x]
%t A343200 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 3], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]
%Y A343200 Cf. A000292, A033488, A219555, A255050, A258343, A338645, A344097, A344098.
%K A343200 nonn
%O A343200 0,2
%A A343200 _Ilya Gutkovskiy_, May 09 2021