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A230256 Expansion of f(-x) * psi(x^2) * phi(x^3) / f(-x^3)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%I A230256 #17 Feb 16 2025 08:33:20
%S A230256 1,-1,0,4,-6,1,11,-19,4,31,-50,11,77,-122,28,173,-273,62,370,-573,130,
%T A230256 751,-1149,261,1461,-2214,498,2750,-4125,923,5022,-7472,1663,8936,
%U A230256 -13202,2919,15551,-22817,5019,26521,-38681,8467,44417,-64438,14035,73197
%N A230256 Expansion of f(-x) * psi(x^2) * phi(x^3) / f(-x^3)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A230256 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A230256 G. C. Greubel, <a href="/A230256/b230256.txt">Table of n, a(n) for n = 0..1000</a>
%H A230256 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A230256 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A230256 Expansion of q^(1/12) * eta(q) * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^5 * eta(q^12)^2) in powers of q.
%F A230256 Euler transform of period 12 sequence [ -1, 0, 4, -2, -1, 0, -1, -2, 4, 0, -1, 0, ...].
%F A230256 a(n) = A132179(2*n) = A062242(4*n) = A062244(4*n) = A132301(4*n) = A182056(4*n) = A182036(6*n) = A182032(12*n - 1).
%F A230256 a(n) = A058531(12*n) = A093073(12*n) = A132976(12*n) = A143840(12*n) = A164268(12*n) = A164612(12*n) = A182033(12*n) = A193261(12*n). - _Michael Somos_, Jan 29 2015
%e A230256 G.f. = 1 - x + 4*x^3 - 6*x^4 + x^5 + 11*x^6 - 19*x^7 + 4*x^8 + 31*x^9 + ...
%e A230256 G.f. = q^-1 - q^11 + 4*q^35 - 6*q^47 + q^59 + 11*q^71 - 19*q^83 + 4*q^95 + ...
%t A230256 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 3, 0, x^3] QPochhammer[ x] / (2 x^(1/4) QPochhammer[ x^3]^3), {x, 0, n}]; (* _Michael Somos_, Jan 29 2015 *)
%o A230256 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^2), n))};
%Y A230256 Cf. A062242, A062244, A132179, A132301, A182032, A182036, A182056.
%Y A230256 Cf. A058531, A093073, A132976, A143840, A164268, A164612, A182033, A193261, A254372.
%K A230256 sign
%O A230256 0,4
%A A230256 _Michael Somos_, Oct 14 2013