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A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

%I A305049 #5 May 24 2018 20:03:54
%S A305049 1,1,3,8,27,67,216,569,1747,4812,14041,39483,115408,326385,941735,
%T A305049 2684170,7725097,22063737,63354066,181223899,519883185,1488316952,
%U A305049 4266788191,12219763777,35023995792,100326757107,287503501905,823654031283,2360146144917,6761847714698,19374935267810
%N A305049 Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).
%C A305049 Invert transform of A163767.
%H A305049 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305049 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A305049 G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).
%p A305049 A:= proc(n, k) option remember; `if`(k=1, 1,
%p A305049       add(A(d, k-1), d=numtheory[divisors](n)))
%p A305049     end:
%p A305049 a:= proc(n) option remember; `if`(n=0, 1,
%p A305049       add(A(j$2)*a(n-j), j=1..n))
%p A305049     end:
%p A305049 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 24 2018
%t A305049 nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t A305049 a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]
%Y A305049 Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.
%K A305049 nonn
%O A305049 0,3
%A A305049 _Ilya Gutkovskiy_, May 24 2018