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A305204 Expansion of Product_{k>=1} 1/(1 - (k*(k + 1)/2)*x^k).

%I A305204 #9 Aug 16 2019 12:52:09
%S A305204 1,1,4,10,29,62,176,363,931,2029,4751,10062,23749,48959,109342,230981,
%T A305204 500344,1031667,2223218,4531585,9570395,19523510,40411313,81628389,
%U A305204 168484616,336850254,685112670,1369559157,2757908932,5464925114,10958578421,21574592680
%N A305204 Expansion of Product_{k>=1} 1/(1 - (k*(k + 1)/2)*x^k).
%H A305204 Alois P. Heinz, <a href="/A305204/b305204.txt">Table of n, a(n) for n = 0..3841</a>
%F A305204 G.f.: Product_{k>=1} 1/(1 - A000217(k)*x^k).
%F A305204 G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j*(j + 1))^k*x^(j*k)/(k*2^k)).
%p A305204 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A305204       1, b(n, i-1)+(1+i)*i/2*b(n-i, min(n-i, i)))
%p A305204     end:
%p A305204 a:= n-> b(n$2):
%p A305204 seq(a(n), n=0..33);  # _Alois P. Heinz_, Aug 16 2019
%t A305204 nmax = 31; CoefficientList[Series[Product[1/(1 - (k (k + 1)/2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A305204 nmax = 31; CoefficientList[Series[Exp[Sum[Sum[(j (j + 1))^k x^(j k)/(k 2^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A305204 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1) ((d + 1)/2)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]
%Y A305204 Cf. A000217, A000294, A006906, A077335, A265836.
%K A305204 nonn
%O A305204 0,3
%A A305204 _Ilya Gutkovskiy_, May 27 2018