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A262156 Expansion of f(-x^6)^3 / (f(x)^2 * psi(x)) in powers of x where psi(), f() are Ramanujan theta functions.

%I A262156 #14 Feb 16 2025 08:33:27
%S A262156 1,-3,8,-19,42,-86,166,-309,557,-974,1661,-2773,4543,-7316,11600,
%T A262156 -18140,28011,-42751,64550,-96503,142951,-209939,305844,-442213,
%U A262156 634865,-905361,1282957,-1807175,2531156,-3526051,4886764,-6739401,9250902,-12641475,17200638
%N A262156 Expansion of f(-x^6)^3 / (f(x)^2 * psi(x)) in powers of x where psi(), f() are Ramanujan theta functions.
%C A262156 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262156 G. C. Greubel, <a href="/A262156/b262156.txt">Table of n, a(n) for n = 0..2500</a>
%H A262156 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262156 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262156 Expansion of q^(-13/24) * eta(q)^3 * eta(q^4)^2 * eta(q^6)^3 / eta(q^2)^8 in powers of q.
%F A262156 Euler transform of period 12 sequence [-3, 5, -3, 3, -3, 2, -3, 3, -3, 5, -3, 0, ...].
%e A262156 G.f. = 1 - 3*x + 8*x^2 - 19*x^3 + 42*x^4 - 86*x^5 + 166*x^6 + ...
%e A262156 G.f. = q^13 - 3*q^37 + 8*q^61 - 19*q^85 + 42*q^109 - 86*q^133 + ...
%t A262156 a[ n_] := SeriesCoefficient[ 2 x^(1/8) QPochhammer[ x^6]^3 / (QPochhammer[ -x]^2 EllipticTheta[ 2 , 0, x^(1/2)]), {x, 0, n}];
%o A262156 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3 / eta(x^2 + A)^8, n))};
%o A262156 (PARI) q='q+O('q^99); Vec(eta(q)^3*eta(q^4)^2*eta(q^6)^3/eta(q^2)^8) \\ _Altug Alkan_, Jul 31 2018
%K A262156 sign
%O A262156 0,2
%A A262156 _Michael Somos_, Sep 13 2015