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A328744 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

%I A328744 #11 Oct 27 2019 13:38:49
%S A328744 1,1,2,2,3,6,5,8,9,13,12,23,18,27,33,39,38,63,54,80,86,101,104,161,
%T A328744 145,183,208,254,256,361,340,435,472,550,600,776,760,918,1018,1221,
%U A328744 1260,1576,1610,1929,2129,2408,2590,3172,3274,3833,4173,4783,5120,6054,6414,7414,8025
%N A328744 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^q(k), where q(k) = number of partitions of k into distinct parts (A000009).
%C A328744 Number of ways to write n as an orderless product of orderless sums with distinct factors and each sum composed of distinct parts. Compare A318949.
%H A328744 Andrew Howroyd, <a href="/A328744/b328744.txt">Table of n, a(n) for n = 1..1000</a>
%e A328744 The a(4) = 2 ways: (4), (3+1).
%e A328744 The a(6) = 6 ways: (6), (4+2), (5+1), (3+2+1), (2)*(3), (2)*(2+1).
%o A328744 (PARI)
%o A328744 MultWeighT(u)={my(n=#u, v=vector(n, k, k==1)); for(k=2, n, if(u[k], my(m=logint(n,k), p=(1 + x + O(x*x^m))^u[k], w=vector(n)); for(i=0, m, w[k^i]=polcoef(p,i)); v=dirmul(v,w))); v}
%o A328744 seq(n)={MultWeighT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1))} \\ _Andrew Howroyd_, Oct 27 2019
%Y A328744 Cf. A000009, A045778, A050368, A318949.
%K A328744 nonn
%O A328744 1,3
%A A328744 _Ilya Gutkovskiy_, Oct 26 2019