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A262158 Expansion of psi(x^3)^3 * psi(x^2) / psi(x)^4 in powers of x where psi() is a Ramanujan theta function.

%I A262158 #14 Jul 23 2025 15:35:28
%S A262158 1,-4,11,-25,53,-107,205,-377,672,-1166,1975,-3275,5333,-8544,13484,
%T A262158 -20994,32288,-49100,73888,-110115,162635,-238196,346123,-499244,
%U A262158 715110,-1017610,1439098,-2023208,2828543,-3933466,5442352,-7493714,10270711,-14014683,19042562
%N A262158 Expansion of psi(x^3)^3 * psi(x^2) / psi(x)^4 in powers of x where psi() is a Ramanujan theta function.
%C A262158 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262158 G. C. Greubel, <a href="/A262158/b262158.txt">Table of n, a(n) for n = 0..2500</a>
%H A262158 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262158 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262158 Expansion of q^(-7/8) * eta(q)^4 * eta(q^4)^2 * eta(q^6)^6 / (eta(q^2)^9 * eta(q^3)^3) in powers of q.
%F A262158 Euler transform of period 12 sequence [-4, 5, -1, 3, -4, 2, -4, 3, -1, 5, -4, 0, ...].
%e A262158 G.f. = 1 - 4*x + 11*x^2 - 25*x^3 + 53*x^4 - 107*x^5 + 205*x^6 + ...
%e A262158 G.f. = q^7 - 4*q^15 + 11*q^23 - 25*q^31 + 53*q^39 - 107*q^47 + ...
%t A262158 a[ n_] := SeriesCoefficient[ x^(-7/8) EllipticTheta[ 2 , 0, x^(3/2)]^3 EllipticTheta[ 2 , 0, x] / EllipticTheta[ 2 , 0, x^(1/2)]^4, {x, 0, n}];
%o A262158 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^2 * eta(x^6 + A)^6 / (eta(x^2 + A)^9 * eta(x^3 + A)^3), n))};
%o A262158 (PARI) q='q+O('q^99); Vec(eta(q)^4*eta(q^4)^2*eta(q^6)^6/(eta(q^2)^9*eta(q^3)^3)) \\ _Altug Alkan_, Jul 31 2018
%K A262158 sign
%O A262158 0,2
%A A262158 _Michael Somos_, Sep 13 2015