A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

1, 2, 3, 5, 6, 8, 11, 13, 16, 19, 22, 26, 30, 34, 38, 44, 49, 54, 62, 67, 74, 83, 89, 98, 107, 115, 124, 134, 145, 155, 168, 178, 189, 206, 217, 231, 247, 259, 277, 294, 310, 327, 345, 365, 382, 404, 424, 444, 470, 489, 513, 539, 561, 588, 613, 641, 670, 699, 729, 756, 791, 824

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

• nonn

Comments

Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).

Examples

			a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Partition Function Q
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.

Programs

Maple

N:= 20: # to get a(0) .. a(A000009(N))
P:= mul(1+x^k,k=1..N):
R:= mul(1+x^coeff(P,x,n)),n=1..N):
seq(coeff(R,x,n),n=0..coeff(P,x,N)); # Robert Israel, Sep 01 2017

Mathematica

nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000009(k)).