A103259 Number of partitions of 2n prime to 3,5 with all odd parts occurring with even multiplicities. There is no restriction on the even parts.

1, 2, 4, 6, 10, 14, 20, 28, 40, 54, 72, 96, 126, 164, 212, 274, 350, 444, 560, 704, 878, 1092, 1352, 1668, 2048, 2506, 3056, 3714, 4500, 5436, 6552, 7872, 9436, 11280, 13456, 16012, 19014, 22532, 26648, 31452, 37052, 43572, 51148, 59940, 70128, 81922, 95548

Offset: 0

Author

Noureddine Chair, Feb 15 2005

Keywords

• nonn

Comments

This is also the sequence A103257/(theta_4(0,x^(15))).

Examples

			a(5) = 14 because 10 can be written as 8+2 = 8+1+1 = 4+4+2 = 4+4+1+1 = 4+2+2+2 = 4+2+2+1+1 = 4+2+1+1+1+1 = 4+1+1+1+1+1+1 = 2+2+2+2+2 = 2+2+2+2+1+1 = 2+2+2+1+1+1+1 = 2+2+1+1+1+1+1+1 = 2+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Crossrefs

Cf. A102346, A103257.

Programs

Maple

series(product((1+x^k)*(1-x^(3*k))*(1-x^(5*k))*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))*(1-x^(15*k))),k=1..100),x=0,100);

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^k)*(1-x^(3*k))*(1-x^(5*k))*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))*(1-x^(15*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

PARI

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)^2*E(5)^2*E(30)) / (E(1)^2*E(6)*E(10)*E(15)^2) ) \\ Joerg Arndt, Sep 01 2015

Formula

G.f.: (theta_4(0, x^3)*theta_4(0, x^5))/(theta_4(0, x)*theta_4(0, x^(15))).

G.f.: (E(2)*E(3)^2*E(5)^2*E(30)) / (E(1)^2*E(6)*E(10)*E(15)^2) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015

a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 01 2015

Extensions

Example corrected by Vaclav Kotesovec, Sep 01 2015

Maple program corrected by Vaclav Kotesovec, Sep 01 2015