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A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A262987 #15 Feb 16 2025 08:33:27
%S A262987 1,1,3,6,11,19,33,53,86,134,205,309,460,672,974,1394,1975,2773,3863,
%T A262987 5333,7316,9964,13484,18140,24269,32288,42751,56331,73888,96503,
%U A262987 125529,162635,209939,270027,346123,442213,563205,715110,905361,1142998,1439098,1807175
%N A262987 Expansion of f(-x, -x^5) * f(x^3, x^5) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%C A262987 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262987 G. C. Greubel, <a href="/A262987/b262987.txt">Table of n, a(n) for n = 0..2500</a>
%H A262987 Vaclav Kotesovec, <a href="https://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 A262987 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262987 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262987 Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
%F A262987 Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].
%F A262987 a(n) = A132217(2*n) = A262160(2*n).
%F A262987 Convolution product of A035294 and A097451.
%F A262987 a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2015
%e A262987 G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
%e A262987 G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...
%t A262987 a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
%t A262987 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jul 31 2018 *)
%o A262987 (PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
%Y A262987 Cf. A035294, A097451, A132217, A262160.
%K A262987 nonn
%O A262987 0,3
%A A262987 _Michael Somos_, Oct 06 2015