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A259033 Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.

%I A259033 #15 Feb 16 2025 08:33:25
%S A259033 1,6,23,76,221,584,1443,3368,7497,16046,33190,66628,130288,248858,
%T A259033 465387,853836,1539425,2731462,4775703,8236856,14027754,23609794,
%U A259033 39301171,64747876,105638153,170778512,273704800,435079524,686237877,1074405242,1670333294,2579418528
%N A259033 Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.
%C A259033 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A259033 Vaclav Kotesovec, <a href="/A259033/b259033.txt">Table of n, a(n) for n = 0..2000</a>
%H A259033 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259033 Expansion of q^(-5/6) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q)^3 * eta(q^3)))^2 in powers of q.
%F A259033 Euler transform of period 6 sequence [ 6, 2, 8, 2, 6, 0, ...].
%F A259033 a(n) = A263528(3*n + 2). -2 * a(n) = A261369(2*n + 1) = A213265(6*n + 5) = A262930(6*n + 5).
%F A259033 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A259033 G.f. = 1 + 6*x + 23*x^2 + 76*x^3 + 221*x^4 + 584*x^5 + 1443*x^6 + 3368*x^7 + ...
%e A259033 G.f. = q^5 + 6*q^11 + 23*q^17 + 76*q^23 + 221*q^29 + 584*q^35 + 1443*q^41 + ...
%t A259033 a[ n_] := SeriesCoefficient[ (1/4) x^(-3/4) EllipticTheta[ 2, 0, x^(3/2)]^2 QPochhammer[ x^2]^4 / QPochhammer[ x]^6, {x, 0, n}];
%t A259033 nmax = 40; CoefficientList[Series[Product[((1 + x^k)^2 * (1 + x^(3*k))^2 * (1 - x^(3*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 16 2017 *)
%o A259033 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A)^3 * eta(x^3 + A)))^2, n))};
%Y A259033 Cf. A213265, A261369, A262930, A263528.
%K A259033 nonn
%O A259033 0,2
%A A259033 _Michael Somos_, Nov 07 2015