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Expansion of Product_{k>=1} (1 + x^k)^tau_k(k), where tau_k(k) = number of ordered k-factorizations of k (A163767).
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%I #15 Nov 03 2018 11:46:05

%S 1,1,2,5,14,22,70,109,318,551,1203,2136,5752,9263,20641,37151,85084,

%T 144918,317356,546730,1196302,2076810,4281584,7459351,15860805,

%U 27146911,54715933,95712097,194059563,334322338,663159101,1147479053,2270647257,3923732160,7587368893

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

%H Seiichi Manyama, <a href="/A321287/b321287.txt">Table of n, a(n) for n = 0..1000</a>

%t tau[n_,1] = 1; tau[n_,k_]:=tau[n,k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 40; CoefficientList[Series[Product[(1+x^k)^tau[k,k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 03 2018, after _Robert G. Wilson v_ *)

%Y Cf. A163767, A304965, A321192.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 02 2018