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A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions.

%I A260412 #12 Feb 16 2025 08:33:26
%S A260412 1,0,2,1,2,2,3,2,3,4,4,5,7,6,9,10,11,12,13,15,17,19,21,24,28,30,35,37,
%T A260412 41,47,52,56,62,69,75,83,92,99,110,121,131,143,157,170,186,203,219,
%U A260412 239,260,281,307,332,359,389,421,453,491,530,570,617,665,714,770
%N A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x where psi(), f(,) are Ramanujan theta functions.
%C A260412 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260412 G. C. Greubel, <a href="/A260412/b260412.txt">Table of n, a(n) for n = 0..2500</a>
%H A260412 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], Sep 30 2015.
%H A260412 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260412 Expansion of q^(1/24) * eta(q^4)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3) * eta(q^12)^2) in powers of q.
%F A260412 Euler transform of period 12 sequence [ 0, 2, 1, -1, 0, 0, 0, -1, 1, 2, 0, -1, ...].
%F A260412 a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - _Vaclav Kotesovec_, Jul 11 2016
%e A260412 G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...
%e A260412 G.f. = 1/q + 2*q^47 + q^71 + 2*q^95 + 2*q^119 + 3*q^143 + 2*q^167 + ...
%t A260412 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 x^(5/8) QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12]), {x, 0, n}];
%t A260412 nmax = 50; CoefficientList[Series[Product[(1-x^(4*k))^3 * (1-x^(6*k))^3 / ((1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 11 2016 *)
%o A260412 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)^2), n))};
%K A260412 nonn
%O A260412 0,3
%A A260412 _Michael Somos_, Jul 24 2015