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A302830 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - k*x^k).

%I A302830 #13 Apr 14 2018 03:45:38
%S A302830 1,2,5,11,25,50,106,203,401,755,1427,2597,4804,8566,15352,27027,47551,
%T A302830 82187,142445,243025,414919,700739,1181236,1972552,3293898,5450728,
%U A302830 9008081,14791741,24244399,39494615,64266141,103979929,167991853,270190879,433773933,693518984
%N A302830 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - k*x^k).
%C A302830 Partial sums of A006906.
%H A302830 Vaclav Kotesovec, <a href="/A302830/b302830.txt">Table of n, a(n) for n = 0..6000</a>
%H A302830 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A302830 G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j).
%F A302830 From _Vaclav Kotesovec_, Apr 14 2018: (Start)
%F A302830 a(n) ~ c * 3^(n/3), where
%F A302830 c = 319343.48587983201292657132469068725642363369445... if mod(n,3)=0
%F A302830 c = 319343.34569378454521307030620964478962032866022... if mod(n,3)=1
%F A302830 c = 319343.21458897980925594955657564398036486423380... if mod(n,3)=2
%F A302830 (End)
%p A302830 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A302830        b(n, i-1)+add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
%p A302830     end:
%p A302830 a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
%p A302830 seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 13 2018
%t A302830 nmax = 35; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A302830 nmax = 35; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%Y A302830 Cf. A000041, A000070, A006906, A091360, A302831.
%K A302830 nonn
%O A302830 0,2
%A A302830 _Ilya Gutkovskiy_, Apr 13 2018