login
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.
2
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
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
Sequence in context: A277277 A241337 A103257 * A280131 A082380 A238871
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