A103260 - OEIS

%I #25 Feb 09 2021 11:04:26

%S 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 174,206,236,266,304,356,420,488,554,624,708,816,946,1084,1224,1372,

%U 1548,1764,2016,2288,2568,2868,3216,3632,4110

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

%C Convolution of A098884 and A003105. [corrected by _Vaclav Kotesovec_, Feb 07 2021]

%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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 Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - _Vladeta Jovovic_, Feb 17 2005

%F a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 01 2015

%F 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 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 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 # alternative program:

%p 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 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 Cf. A003105, A098884, A080054, A001082, A098151.

%K nonn,easy

%O 0,2

%A _Noureddine Chair_, Feb 15 2005

%E Example corrected by _Vaclav Kotesovec_, Sep 01 2015