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A208435 Expansion of x * f(x) * f(-x^12)^3 * psi(x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

%I A208435 #14 Feb 16 2025 08:33:16
%S A208435 0,1,1,-2,0,3,-2,-4,0,5,1,-8,0,7,-4,-8,0,9,8,-10,0,14,-6,-12,0,16,6,
%T A208435 -14,0,15,-8,-20,0,17,14,-18,0,19,-10,-24,0,26,1,-22,0,23,-16,-28,0,
%U A208435 25,20,-32,0,32,-14,-28,0,29,12,-30,0,38,-16,-32,0,33,31
%N A208435 Expansion of x * f(x) * f(-x^12)^3 * psi(x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.
%C A208435 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A208435 G. C. Greubel, <a href="/A208435/b208435.txt">Table of n, a(n) for n = 0..1000</a>
%H A208435 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208435 Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6)^2 * eta(q^12)^3 / (eta(q) * eta(q^3) * eta(q^4)^3) in powers of q.
%F A208435 Euler transform of period 12 sequence [ 1, -3, 2, 0, 1, -4, 1, 0, 2, -3, 1, -4, ...].
%F A208435 a(4*n) = 0. 24 * a(n) = A207541(3*n + 2).
%e A208435 x + x^2 - 2*x^3 + 3*x^5 - 2*x^6 - 4*x^7 + 5*x^9 + x^10 - 8*x^11 + ...
%e A208435 q^5 + q^8 - 2*q^11 + 3*q^17 - 2*q^20 - 4*q^23 + 5*q^29 + q^32 - 8*q^35 + ...
%t A208435 a[n_]:=SeriesCoefficient[((QP[q^2]^4*QP[q^6]^2*QP[q^12]^3)/(QP[q]*QP[q^3]*
%t A208435 QP[q^4]^3)), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* _G. C. Greubel_, Dec 17 2017 *)
%o A208435 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^3), n))}
%Y A208435 Cf. A207541.
%K A208435 sign
%O A208435 0,4
%A A208435 _Michael Somos_, Feb 26 2012