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A320652 Expansion of 1/(2 - Product_{k>=1} 1/(1 - k*x^k)).

%I A320652 #6 Apr 02 2019 05:52:36
%S A320652 1,1,4,13,45,147,497,1643,5490,18252,60812,202364,673915,2243295,
%T A320652 7468973,24865272,82783967,275605513,917563193,3054785032,10170143277,
%U A320652 33858882922,112724577088,375287739083,1249425198725,4159643200494,13848474406054,46104972636634,153494780854254
%N A320652 Expansion of 1/(2 - Product_{k>=1} 1/(1 - k*x^k)).
%C A320652 Invert transform of A006906.
%H A320652 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320652 G.f.: 1/(1 - Sum_{k>=1} k*x^k / Product_{j=1..k} (1 - j*x^j)).
%F A320652 a(0) = 1; a(n) = Sum_{k=1..n} A006906(k)*a(n-k).
%p A320652 a:=series(1/(2-mul(1/(1-k*x^k),k=1..100)),x=0,29): seq(coeff(a,x,n),n=0..28); # _Paolo P. Lava_, Apr 02 2019
%t A320652 nmax = 28; CoefficientList[Series[1/(2 - Product[1/(1 - k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
%t A320652 nmax = 28; CoefficientList[Series[1/(1 - Sum[k x^k/Product[(1 - j x^j), {j, 1, k}], {k, 1, nmax}]), {x, 0, nmax}], x]
%t A320652 a[0] = 1; a[n_] := a[n] = Sum[Total[Times@@@IntegerPartitions[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]
%Y A320652 Cf. A006906, A055887, A257674, A299162.
%K A320652 nonn
%O A320652 0,3
%A A320652 _Ilya Gutkovskiy_, Oct 18 2018