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A228447 Expansion of q * (psi(q^3) * psi(q^6)) / (psi(q) * phi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A228447 #14 Feb 16 2025 08:33:20
%S A228447 1,-3,7,-15,30,-57,104,-183,313,-522,852,-1365,2150,-3336,5106,-7719,
%T A228447 11538,-17067,25004,-36306,52280,-74700,105960,-149277,208951,-290706,
%U A228447 402127,-553224,757158,-1031166,1397744,-1886151,2534316,-3391254,4520112,-6002007
%N A228447 Expansion of q * (psi(q^3) * psi(q^6)) / (psi(q) * phi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A228447 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A228447 G. C. Greubel, <a href="/A228447/b228447.txt">Table of n, a(n) for n = 1..1000</a>
%H A228447 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A228447 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A228447 Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(q) * phi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A228447 Expansion of eta(q)^3 * eta(x^4)^2 * eta(x^6) * eta(x^12)^2 / (eta(x^2)^7 * eta(x^3)) in powers of q.
%F A228447 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - 3*v) - v * (1 - 4*v) * (1 - 3*u)^2.
%F A228447 a(n) = -(-1)^n * A187100(n). a(2*n) = -3 * A128638(n).
%F A228447 Convolution inverse is A187145. Convolution with A033716 is A093829.
%e A228447 G.f. = q - 3*q^2 + 7*q^3 - 15*q^4 + 30*q^5 - 57*q^6 + 104*q^7 - 183*q^8 + ...
%t A228447 a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3]), {q, 0, n}]
%o A228447 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^7 * eta(x^3 + A)), n))}
%Y A228447 Cf. A033716, A093829, A128638, A187100, A187145.
%K A228447 sign
%O A228447 1,2
%A A228447 _Michael Somos_, Oct 26 2013