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A262162 Expansion of f(-x^2)^5 * f(-x^12)^3 / (f(x)^2 * f(-x^8)^6) in powers of x where f() is a Ramanujan theta function.

%I A262162 #10 Feb 16 2025 08:33:27
%S A262162 1,-2,0,0,0,4,0,0,1,-12,0,0,-3,30,0,0,4,-66,0,0,-3,136,0,0,5,-268,0,0,
%T A262162 -12,506,0,0,14,-920,0,0,-10,1622,0,0,18,-2788,0,0,-37,4688,0,0,41,
%U A262162 -7726,0,0,-34,12506,0,0,54,-19928,0,0,-98,31306,0,0,109
%N A262162 Expansion of f(-x^2)^5 * f(-x^12)^3 / (f(x)^2 * f(-x^8)^6) in powers of x where f() is a Ramanujan theta function.
%C A262162 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262162 G. C. Greubel, <a href="/A262162/b262162.txt">Table of n, a(n) for n = 0..1000</a>
%H A262162 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262162 Expansion of q^(1/6) * eta(q)^2 * eta(q^4)^2 * eta(q^12)^3 / (eta(q^2) * eta(q^8)^6) in powers of q.
%F A262162 Euler transform of period 24 sequence [ -2, -1, -2, -3, -2, -1, -2, 3, -2, -1, -2, -6, -2, -1, -2, 3, -2, -1, -2, -3, -2, -1, -2, 0, ...].
%F A262162 a(4*n) = A262150(n). a(4*n + 1) = -2 * A262151(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A262162 G.f. = 1 - 2*x + 4*x^5 + x^8 - 12*x^9 - 3*x^12 + 30*x^13 + 4*x^16 + ...
%e A262162 G.f. = q^-1 - 2*q^5 + 4*q^29 + q^47 - 12*q^53 - 3*q^71 + 30*q^77 + ...
%t A262162 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 QPochhammer[ x^12]^3 / (QPochhammer[ -x]^2 QPochhammer[ x^8]^6), {x, 0, n}];
%o A262162 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A)^3 / (eta(x^2 + A) * eta(x^8 + A)^6), n))};
%Y A262162 Cf. A262150, A262151.
%K A262162 sign
%O A262162 0,2
%A A262162 _Michael Somos_, Sep 13 2015