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%I #18 Mar 12 2021 22:24:42
%S 1,2,5,10,20,36,65,110,185,300,481,754,1169,1780,2685,3996,5894,8600,
%T 12450,17860,25442,35964,50519,70490,97800,134892,185099,252664,
%U 343280,464200,625033,837998,1119114,1488720,1973210,2606028,3430238,4500224,5885540
%N Expansion of 1 / Product_{k >= 1} (1-q^k)^2*(1-q^(11k))^2.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A032442/b032442.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of 1 / (f(-x) * f(-x^11))^2 in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Apr 21 2015
%F Expansion of q / eta(q)^2 * eta(q^11)^2 in powers of q. - _Michael Somos_, Apr 21 2015
%F Euler transform of period 11 sequence [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...]. - _Michael Somos_, Apr 21 2015
%F Given g.f. A(x), then B(q) = A(q)/q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (w^2 + 16*v^2) - v^2 * (v + 4*u) * (w + 4*u). - _Michael Somos_, Apr 21 2015
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-1 (t/i)^-2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Apr 21 2015
%F G.f.: (Product_{k > 0} (1 - x^k)^2 * (1 - x^(11*k)))^-2.
%F Convolution inverse of A006571. Convolution with A028610 is A128525. - _Michael Somos_, Apr 21 2015
%F a(n) ~ exp(4*Pi*sqrt(n/11)) / (sqrt(2) * 11^(1/4) * n^(7/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + ...
%e G.f. = 1/q + 2 + 5*q + 10*q^2 + 20*q^3 + 36*q^4 + 65*q^5 + 110*q^6 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^-2, {x, 0, n}]; (* _Michael Somos_, Apr 21 2015 *)
%t nmax=60; CoefficientList[Series[Product[1/((1-x^k)^2 * (1-x^(11*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^-2, n))}; /* _Michael Somos_, Apr 21 2015 */
%Y Cf. A006571, A028610, A128525.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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