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%I #10 Mar 12 2021 22:24:46
%S 1,1,0,0,-1,0,0,0,0,-2,0,0,4,4,0,0,-6,-1,0,0,1,-8,0,0,11,14,0,0,-19,
%T -4,0,0,4,-23,0,0,31,40,0,0,-50,-10,0,0,11,-60,0,0,77,98,0,0,-122,-24,
%U 0,0,28,-140,0,0,173,224,0,0,-273,-54,0,0,62,-304,0,0
%N Expansion of psi(x) * chi(-x^3) * f(-x^16) * chi(-x^24) / phi(-x^12)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A182056/b182056.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/3) * eta(q^2)^2 * eta(q^3) * eta(q^16) * eta(q^24)^3 / (eta(q) * eta(q^6) * eta(q^12)^4 * eta(q^48)) in powers of q.
%F Euler transform of period 48 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 0, -1, 1, 3, 1, -1, 0, -2, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, -2, 0, -1, 1, 3, 1, -1, 0, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].
%F a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A182032(12*n - 1). a(4*n + 1) = A182032(12*n + 2).
%e 1 + x - x^4 - 2*x^9 + 4*x^12 + 4*x^13 - 6*x^16 - x^17 + x^20 + ...
%e 1/q + q^2 - q^11 - 2*q^26 + 4*q^35 + 4*q^38 - 6*q^47 - q^50 + q^59 + ...
%t QP := QPochhammer; A182056[n_] := SeriesCoefficient[QP[q^2]^2*QP[q^3]* QP[q^16]*QP[q^24]^3/(QP[q]* QP[q^6]*QP[q^12]^4*QP[q^48]), {q, 0, n}];
%t Table[A182056[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 24 2017 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^16 + A) * eta(x^24 + A)^3 / (eta(x + A) * eta(x^6 + A) * eta(x^12 + A)^4 * eta(x^48 + A)), n))}
%Y Cf. A182032.
%K sign
%O 0,10
%A _Michael Somos_, Apr 08 2012
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