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A260150 Expansion of f(x, x^5)^3 / (f(-x, -x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.

%I A260150 #16 Feb 16 2025 08:33:26
%S A260150 1,4,11,24,48,92,170,304,526,884,1451,2336,3700,5772,8876,13472,20207,
%T A260150 29988,44072,64184,92680,132760,188758,266512,373838,521152,722266,
%U A260150 995432,1364684,1861548,2527224,3415344,4595497,6157700,8218050,10925848,14472520
%N A260150 Expansion of f(x, x^5)^3 / (f(-x, -x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.
%C A260150 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260150 Vaclav Kotesovec, <a href="/A260150/b260150.txt">Table of n, a(n) for n = 0..2000</a>
%H A260150 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260150 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260150 Expansion of f(x) * psi(-x^3)^3 / (f(-x)^3 * psi(x^3)) in powers of x where psi(), f() are Ramanujan theta functions.
%F A260150 Euler transform of period 12 sequence [ 4, 1, 0, 2, 4, 2, 4, 2, 0, 1, 4, 0, ...].
%F A260150 a(n) = (-1)^n * A260057(n). a(n) = A261154(3*n + 2). a(2*n + 1) = 4 * A259033(n).
%F A260150 a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A260150 G.f. = 1 + 4*x + 11*x^2 + 24*x^3 + 48*x^4 + 92*x^5 + 170*x^6 + 304*x^7 + ...
%e A260150 G.f. = q^2 + 4*q^5 + 11*q^8 + 24*q^11 + 48*q^14 + 92*q^17 + 170*q^20 + ...
%t A260150 a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) QPochhammer[ -x] / QPochhammer[x]^3 EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}];
%o A260150 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^12 + A)^3 / (eta(x + A)^4 * eta(x^4 + A) * eta(x^6 + A)^5), n))};
%Y A260150 Cf. A259033, A260057, A261154.
%K A260150 nonn
%O A260150 0,2
%A A260150 _Michael Somos_, Nov 08 2015