cp's OEIS Frontend

A103257 Number of partitions of 2n free of multiples of 5. All odd parts occur with multiplicity 2 or 4. the even parts occur at most twice.

%I A103257 #26 Aug 31 2024 23:28:42
%S A103257 1,2,4,6,10,14,20,28,40,54,72,96,126,164,212,272,346,436,548,684,850,
%T A103257 1052,1296,1588,1940,2362,2864,3462,4172,5012,6004,7172,8548,10160,
%U A103257 12048,14256,16830,19828,23312,27356,32040
%N A103257 Number of partitions of 2n free of multiples of 5. All odd parts occur with multiplicity 2 or 4. the even parts occur at most twice.
%C A103257 Convolution of A261796 and A261797. - _Vaclav Kotesovec_, Sep 01 2015
%H A103257 Robert Israel, <a href="/A103257/b103257.txt">Table of n, a(n) for n = 0..10000</a>
%H A103257 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.
%F A103257 G.f.: (theta_4(0, x^3)*theta_4(0, x^5))/theta_4(0, x).
%F A103257 G.f.: (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) where E(k) = Product_{n>=1} 1-q^(k*n). - _Joerg Arndt_, Sep 01 2015
%F A103257 a(n) ~ exp(Pi*sqrt(7*n/15)) / sqrt(15*n). - _Vaclav Kotesovec_, Sep 01 2015
%e A103257 E.g. a(5) = 14 because 10 can be written as 8+2 = 8+1+1 = 6+4 = 6+2+2 = 6+2+1+1 = 6+1+1+1+1 = 4+4+2 = 4+4+1+1 = 4+3+3 = 4+2+2+1+1 = 4+2+1+1+1+1 = 3+3+2+2 = 3+3+2+1+1 = 3+3+1+1+1+1.
%p A103257 series(product(((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);
%t A103257 nmax = 50; CoefficientList[Series[Product[((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 01 2015 *)
%o A103257 (PARI)  q='q+O('q^33); E(k)=eta(q^k);
%o A103257 Vec( (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) ) \\ _Joerg Arndt_, Sep 01 2015
%Y A103257 Cf. A098151, A261796, A261797.
%K A103257 nonn
%O A103257 0,2
%A A103257 _Noureddine Chair_, Jan 27 2005
%E A103257 Example corrected by _Vaclav Kotesovec_, Sep 01 2015
%E A103257 Maple program fixed by _Vaclav Kotesovec_, Sep 01 2015