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

0,2

Comments

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

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A102346, A103257.

Sequence in context: A277277 A241337 A103257 * A280131 A082380 A238871

Adjacent sequences: A103256 A103257 A103258 * A103260 A103261 A103262

Keyword

nonn

Author

Noureddine Chair, Feb 15 2005

Extensions

Example corrected by Vaclav Kotesovec, Sep 01 2015

Maple program corrected by Vaclav Kotesovec, Sep 01 2015

Status

approved