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A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

%I A318127 #8 Apr 03 2019 02:59:17
%S A318127 1,2,6,19,61,191,588,1785,5351,15868,46628,135921,393318,1130538,
%T A318127 3229753,9175347,25931605,72936434,204223348,569427145,1581458917,
%U A318127 4375905243,12065914843,33160240020,90848002909,248154744196,675932128695,1836182233332,4975249827916,13447775233746
%N A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).
%C A318127 Binomial transform of A006906.
%H A318127 Vaclav Kotesovec, <a href="/A318127/b318127.txt">Table of n, a(n) for n = 0..2000</a>
%H A318127 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A318127 G.f.: (1/(1 - x))*exp(Sum_{k>=1} Sum_{j>=1} j^k*x^(k*j)/(k*(1 - x)^(k*j))).
%F A318127 a(n) = Sum_{k=0..n} binomial(n,k)*A006906(k).
%F A318127 a(n) ~ c * (1 + 3^(1/3))^n, where c = 97923.037496367052161042295948902147352859984491653037730624387144966464... = 1/((3^(1/3) - 1) * (3^(2/3) - 2)) * Product_{k>=4} 1/(1 - k/3^(k/3)). - _Vaclav Kotesovec_, Aug 19 2018
%p A318127 a:=series(1/(1-x)*mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,30): seq(coeff(a,x,n),n=0..29); # _Paolo P. Lava_, Apr 02 2019
%t A318127 nmax = 29; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A318127 nmax = 29; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[j^k x^(k j)/(k (1 - x)^(k j)), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A318127 Table[Sum[Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 29}]
%Y A318127 Cf. A006906, A218481, A294500.
%K A318127 nonn
%O A318127 0,2
%A A318127 _Ilya Gutkovskiy_, Aug 18 2018