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A210656 Expansion of psi(x^3) * phi(-x)^2 / phi(-x^2) in power of x where phi(), psi() are Ramanujan theta functions.

%I A210656 #14 Feb 16 2025 08:33:17
%S A210656 1,8,36,130,412,1176,3105,7712,18192,41098,89476,188592,386322,771528,
%T A210656 1506036,2879688,5403628,9966408,18092599,32366288,57117660,99526362,
%U A210656 171378512,291841464,491812740,820684904,1356794820,2223458146,3613417008,5825889936
%N A210656 Expansion of psi(x^3) * phi(-x)^2 / phi(-x^2) in power of x where phi(), psi() are Ramanujan theta functions.
%C A210656 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210656 Vaclav Kotesovec, <a href="/A210656/b210656.txt">Table of n, a(n) for n = 0..2000</a>
%H A210656 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210656 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210656 Expansion of q^(-3/4) * ( eta(q^2)^4 * eta(q^6)^2 / (eta(q)^4 * eta(q^3) * eta(q^ 4)) )^2 in powers of q.
%F A210656 Euler transform of period 12 sequence [ 8, 0, 10, 2, 8, -2, 8, 2, 10, 0, 8, 0, ...].
%F A210656 A001936(9*n + 2) - A001936(n) = 4 * a(3*n). A001936(9*n + 5) = 4 * a(3*n + 1). A001936(9*n + 8) = 4 * a(3*n + 2).
%F A210656 a(n) ~ exp(sqrt(3*n)*Pi) / (32*sqrt(2)*3^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A210656 1 + 8*x + 36*x^2 + 130*x^3 + 412*x^4 + 1176*x^5 + 3105*x^6 + 7712*x^7 + ...
%e A210656 q^3 + 8*q^7 + 36*q^11 + 130*q^15 + 412*q^19 + 1176*q^23 + 3105*q^27 + ...
%t A210656 nmax = 30; CoefficientList[Series[Product[((1 - x^(2*k))^4 * (1 - x^(6*k))^2 / ((1 - x^k)^4 * (1 - x^(3*k)) * (1 - x^(4*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 16 2017 *)
%o A210656 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A)^4 * eta(x^3 + A) * eta(x^ 4 + A)) )^2, n))}
%Y A210656 Cf. A001936.
%K A210656 nonn
%O A210656 0,2
%A A210656 _Michael Somos_, Mar 27 2012