cp's OEIS Frontend

A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

%I A304965 #9 Nov 02 2018 10:36:12
%S A304965 1,1,3,6,19,30,96,152,461,775,1883,3271,8751,14370,34004,59491,140450,
%T A304965 239746,541817,932681,2089189,3606641,7719178,13398411,28848808,
%U A304965 49603982,103047935,179154858,370200348,639269735,1295389370,2241994088,4511677298,7798101800,15408901600
%N A304965 Expansion of Product_{k>=1} 1/(1 - x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).
%C A304965 Euler transform of A163767.
%H A304965 Seiichi Manyama, <a href="/A304965/b304965.txt">Table of n, a(n) for n = 0..1000</a>
%H A304965 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A304965 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A304965 G.f.: Product_{k>=1} 1/(1 - x^k)^A163767(k).
%p A304965 A:= proc(n, k) option remember; `if`(k=1, 1,
%p A304965       add(A(d, k-1), d=numtheory[divisors](n)))
%p A304965     end:
%p A304965 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A304965       A(d$2), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A304965     end:
%p A304965 seq(a(n), n=0..40);  # _Alois P. Heinz_, May 22 2018
%t A304965 nmax = 34; CoefficientList[Series[Product[1/(1 - x^k)^Times@@(Binomial[# + k - 1, k - 1]&/@FactorInteger[k][[All, 2]]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A304965 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Times@@(Binomial[# + d - 1, d - 1]&/@FactorInteger[d][[All, 2]]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]
%Y A304965 Cf. A000219, A001970, A006171, A129373, A129374, A163767, A174465, A280487.
%K A304965 nonn
%O A304965 0,3
%A A304965 _Ilya Gutkovskiy_, May 22 2018