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

1, 1, 2, 2, 3, 6, 5, 8, 9, 13, 12, 23, 18, 27, 33, 39, 38, 63, 54, 80, 86, 101, 104, 161, 145, 183, 208, 254, 256, 361, 340, 435, 472, 550, 600, 776, 760, 918, 1018, 1221, 1260, 1576, 1610, 1929, 2129, 2408, 2590, 3172, 3274, 3833, 4173, 4783, 5120, 6054, 6414, 7414, 8025

Offset: 1

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

• nonn

Comments

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.

Examples

			The a(4) = 2 ways: (4), (3+1).
The a(6) = 6 ways: (6), (4+2), (5+1), (3+2+1), (2)*(3), (2)*(2+1).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000009, A045778, A050368, A318949.

Programs

PARI

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}
seq(n)={MultWeighT(Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1))} \\ Andrew Howroyd, Oct 27 2019