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A254346 Expansion of f(x, x^5) * f(-x^6) / f(x)^2 in powers of x where f() is a Ramanujan theta function.

%I A254346 #11 Feb 16 2025 08:33:24
%S A254346 1,-1,3,-5,10,-15,26,-39,63,-92,140,-201,295,-415,591,-818,1140,-1554,
%T A254346 2126,-2861,3855,-5126,6816,-8970,11793,-15372,20007,-25857,33356,
%U A254346 -42771,54734,-69683,88530,-111968,141312,-177642,222842,-278557,347484,-432095,536230
%N A254346 Expansion of f(x, x^5) * f(-x^6) / f(x)^2 in powers of x where f() is a Ramanujan theta function.
%C A254346 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A254346 G. C. Greubel, <a href="/A254346/b254346.txt">Table of n, a(n) for n = 0..1000</a>
%H A254346 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A254346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A254346 Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^2)^4 in powers of q.
%F A254346 Euler transform of period 12 sequence [ -1, 3, -2, 2, -1, 2, -1, 2, -2, 3, -1, 0, ...].
%F A254346 a(n) = (-1)^n * A132302(n). 2 * a(n) = A254372(2*n + 1).
%e A254346 G.f. = 1 - x + 3*x^2 - 5*x^3 + 10*x^4 - 15*x^5 + 26*x^6 - 39*x^7 + ...
%e A254346 G.f. = q - q^3 + 3*q^5 - 5*q^7 + 10*q^9 - 15*q^11 + 26*q^13 - 39*q^15 + ...
%t A254346 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^12] / (QPochhammer[ x^2] QPochhammer[ -x]), {x, 0, n}];
%o A254346 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^2 + A)^4, n))};
%Y A254346 Cf. A132302, A254372.
%K A254346 sign
%O A254346 0,3
%A A254346 _Michael Somos_, Jan 29 2015