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A261877 Expansion of psi(x^4) / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A261877 #21 Feb 16 2025 08:33:27
%S A261877 1,0,0,2,1,0,4,2,0,8,4,0,15,8,0,26,14,0,44,24,0,72,40,0,115,64,0,180,
%T A261877 100,0,276,154,0,416,232,0,618,344,0,906,505,0,1312,730,0,1880,1044,0,
%U A261877 2666,1480,0,3746,2076,0,5220,2888,0,7216,3988,0,9903,5464,0
%N A261877 Expansion of psi(x^4) / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A261877 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261877 G. C. Greubel, <a href="/A261877/b261877.txt">Table of n, a(n) for n = 0..2500</a>
%H A261877 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261877 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261877 Expansion of q^(-1/2) * eta(q^6) * eta(q^8)^2 / (eta(q^3)^2 * eta(q^4)) in powers of q.
%F A261877 Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, 1, 0, -1, 2, 0, 0, 2, 0, 0, 2, -1, 0, 1, 0, 1, 2, 0, 0, 0, ...].
%F A261877 2 * a(n) = A143068(2*n + 1). a(3*n + 2) = 0.
%F A261877 Convolution inverse is A262929. - _Michael Somos_, Oct 22 2017
%e A261877 G.f. = 1 + 2*x^3 + x^4 + 4*x^6 + 2*x^7 + 8*x^9 + 4*x^10 + 15*x^12 + ...
%e A261877 G.f. = q + 2*q^7 + q^9 + 4*q^13 + 2*q^15 + 8*q^19 + 4*q^21 + 15*q^25 + ...
%t A261877 a[ n_] := SeriesCoefficient[ 1/2 (x^2)^(-1/4) EllipticTheta[ 2, 0, x^2] / EllipticTheta[ 4, 0, x^3], {x, 0, n}];
%o A261877 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^8 + A)^2 / (eta(x^3 + A)^2 * eta(x^4 + A)), n))};
%Y A261877 Cf. A143068, A262929.
%K A261877 nonn
%O A261877 0,4
%A A261877 _Michael Somos_, Sep 09 2015