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A260518 Expansion of psi(x)^2 * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%I A260518 #14 Jul 14 2025 22:52:07
%S A260518 1,3,5,7,8,11,13,14,17,16,21,23,25,27,21,32,33,35,37,32,42,38,45,47,
%T A260518 40,51,56,55,50,48,61,63,64,70,56,62,73,80,77,63,81,83,74,87,72,91,98,
%U A260518 95,104,64,101,103,105,107,88,112,98,115,114,112,121,123,125
%N A260518 Expansion of psi(x)^2 * f(-x^3)^3 / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.
%C A260518 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260518 G. C. Greubel, <a href="/A260518/b260518.txt">Table of n, a(n) for n = 0..2500</a>
%H A260518 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260518 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260518 Expansion of psi(x) * psi(x^3) * f(x, x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.
%F A260518 Expansion of q^(-7/12) * eta(q^2)^4 * eta(q^3)^3 / eta(q)^3 in powers of q.
%F A260518 Euler transform of period 6 sequence [ 3, -1, 0, -1, 3, -4, ...].
%F A260518 G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k + x^(2*k))^3.
%e A260518 G.f. = 1 + 3*x + 5*x^2 + 7*x^3 + 8*x^4 + 11*x^5 + 13*x^6 + 14*x^7 + ...
%e A260518 G.f. = q^7 + 3*q^19 + 5*q^31 + 7*q^43 + 8*q^55 + 11*q^67 + 13*q^79 + ...
%t A260518 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 QPochhammer[ x^3]^3 / QPochhammer[ x]^3, {x, 0, n}];
%o A260518 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^3 / eta(x + A)^3, n))};
%o A260518 (PARI) my(q='q+O('q^99)); Vec(eta(q^2)^4*eta(q^3)^3/eta(q)^3) \\ _Altug Alkan_, Aug 01 2018
%K A260518 nonn
%O A260518 0,2
%A A260518 _Michael Somos_, Jul 28 2015