%I #14 Dec 11 2016 01:53:47
%S 1,2,3,5,8,12,18,26,37,52,72,99,134,180,240,317,416,542,702,904,1158,
%T 1476,1872,2364,2973,3724,4647,5778,7160,8844,10890,13370,16368,19984,
%U 24336,29561,35822,43308,52242,62884,75536,90552,108342,129384,154232
%N G.f.: q*Product_{k>0} (1-q^(12k))(1+q^(12k-1))(1+q^(12k-11))/(1-q^k).
%C Coefficients of a q-series of Andrews inspired by Ramanujan.
%H G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%F Euler transform of period 24 sequence [ 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, ...]. - _Michael Somos_, Sep 19 2006
%F Given g.f. A(x), then B(x)=A(x)+A(x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+6*u)*v*(1+2*v)-u^2. - _Michael Somos_, Sep 19 2006
%F G.f.: q*{Sum_{k} q^(24k^2+10k) +q^(24k^2+14k+1) }/{Sum_{k} (-1)^k q^((3k^2+k)/2) }. - _Michael Somos_, Sep 19 2006
%F G.f.: q*Product_{k>0} (1-q^(12k))(1+q^(12k-1))(1+q^(12k-11))/(1-q^k).
%F G.f.: Sum_{k>0} Prod[i=1..k, (1+q^i)^2]*(1+q^k)*q^(k^2) /{(1-q)(1-q^2)...(1-q^(2k))}.
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 31 2015
%t nmax = 100; Rest[CoefficientList[Series[x*Product[(1-x^(12*k)) * (1+x^(12*k-1)) * (1+x^(12*k-11))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%o (PARI) {a(n)=if(n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2/(1+x^k)* prod(i=1, k, (1+x^i)^2/(1-x^(2*i-1))/(1-x^(2*i)), 1+x*O(x^(n-k^2)))), n))} /* _Michael Somos_, Sep 19 2006 */
%Y Cf. A036018.
%K nonn
%O 1,2
%A _Ralf Stephan_, Sep 21 2004