cp's OEIS Frontend

A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

%I A103260 #33 Feb 16 2025 08:32:56
%S A103260 1,2,2,2,2,4,6,8,10,10,12,16,22,28,32,36,42,52,66,80,92,104,120,144,
%T A103260 174,206,236,266,304,356,420,488,554,624,708,816,946,1084,1224,1372,
%U A103260 1548,1764,2016,2288,2568,2868,3216,3632,4110,4626,5166,5748,6412,7188
%N A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.
%C A103260 Convolution of A098884 and A003105. [corrected by _Vaclav Kotesovec_, Feb 07 2021]
%C A103260 Also equal to the number of overpartitions of n into parts congruent to 1 or 5 modulo 6. - _Jeremy Lovejoy_, Nov 28 2024
%H A103260 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.
%H A103260 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A103260 G.f.: (Theta_4(0, x^2)*theta_4(0, x^3))/(theta_4(0, x)*theta_4(0, x^(6))) = Product_{k>0}((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))).
%F A103260 Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - _Vladeta Jovovic_, Feb 17 2005
%F A103260 a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 01 2015
%F A103260 G.f.: f(x,x^5)/f(-x,-x^5) = ( Sum_{n = -oo..oo} x^(n*(3*n-2)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(3*n-2)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. Cf. A080054 and A098151. - _Peter Bala_, Feb 05 2021
%e A103260 E.g. a(7)=8 because 14=10+4=10+2+1+1=8+4+2=8+4+1+1=7+7=5+5+4=5+5+2+1+1.
%p A103260 series(product(((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))),k=1..100),x=0,100);
%p A103260 # alternative program:
%p A103260 with(gfun): series( add(x^(n*(3*n-2)), n = -6..6)/add((-1)^n*x^(n*(3*n-2)), n = -6..6), x, 100): seriestolist(%); # _Peter Bala_, Feb 05 2021
%t A103260 nmax = 50; CoefficientList[Series[Product[((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 01 2015 *)
%Y A103260 Cf. A003105, A098884, A080054, A001082, A098151, A015128.
%K A103260 nonn,easy
%O A103260 0,2
%A A103260 _Noureddine Chair_, Feb 15 2005
%E A103260 Example corrected by _Vaclav Kotesovec_, Sep 01 2015