cp's OEIS Frontend

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.

%I A103259 #23 Sep 01 2015 19:43:29
%S A103259 1,2,4,6,10,14,20,28,40,54,72,96,126,164,212,274,350,444,560,704,878,
%T A103259 1092,1352,1668,2048,2506,3056,3714,4500,5436,6552,7872,9436,11280,
%U A103259 13456,16012,19014,22532,26648,31452,37052,43572,51148,59940,70128,81922,95548
%N 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.
%C A103259 This is also the sequence A103257/(theta_4(0,x^(15))).
%H A103259 Alois P. Heinz, <a href="/A103259/b103259.txt">Table of n, a(n) for n = 0..10000</a>
%H A103259 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.
%F A103259 G.f.: (theta_4(0, x^3)*theta_4(0, x^5))/(theta_4(0, x)*theta_4(0, x^(15))).
%F A103259 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
%F A103259 a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 01 2015
%e A103259 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.
%p A103259 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);
%t A103259 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 *)
%o A103259 (PARI)  q='q+O('q^33); E(k)=eta(q^k);
%o A103259 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
%Y A103259 Cf. A102346, A103257.
%K A103259 nonn
%O A103259 0,2
%A A103259 _Noureddine Chair_, Feb 15 2005
%E A103259 Example corrected by _Vaclav Kotesovec_, Sep 01 2015
%E A103259 Maple program corrected by _Vaclav Kotesovec_, Sep 01 2015