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%I #16 Sep 08 2022 08:46:12
%S 1,1,-2,0,1,-4,0,0,-2,4,2,0,0,2,0,0,1,-4,4,0,-4,0,0,0,0,3,-4,0,0,-4,0,
%T 0,-2,0,2,0,4,2,0,0,2,-4,0,0,0,8,0,0,0,1,-6,0,2,-4,0,0,0,0,2,0,0,2,0,
%U 0,1,-8,0,0,-4,0,0,0,4,2,-4,0,0,0,0,0,-4,4
%N Expansion of f(q) * f(-q^2) * chi(q^3) in powers of q where 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="/A258228/b258228.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 f(q)^2 * f(-q^6) / f(q, q^5) in powers of q where f(,) is Ramanujan's general theta function.
%F Expansion of eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [ 1, -3, 2, -2, 1, -4, 1, -2, 2, -3, 1, -2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122865.
%F G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 * (1 + x^(3*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))).
%F a(n) = (-1)^n * A258210(n) = A258279(2*n) = A258292(2*n).
%F a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0.
%F a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(12*n + 1) = A002175(n).
%e G.f. = 1 + q - 2*q^2 + q^4 - 4*q^5 - 2*q^8 + 4*q^9 + 2*q^10 + 2*q^13 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^2 / (QPochhammer[ -q, q^6] QPochhammer[ -q^5, q^6]), {q, 0, n}];
%o (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] + A[2] - 2*A[3] + A[5] - 4*A[6] - 2*A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] - 4*A[18] + 4*A[19];
%Y Cf. A002175, A004018, A122856, A122865, A258210, A258279, A258292.
%K sign
%O 0,3
%A _Michael Somos_, May 23 2015
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