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A261203 Expansion of f(-x^6)^2 / (phi(-x) * phi(-x^9)) in powers of x where phi(), f() are Ramanujan theta functions.

%I A261203 #12 Feb 16 2025 08:33:26
%S A261203 1,2,4,8,14,24,38,60,92,140,208,304,439,626,884,1232,1704,2336,3182,
%T A261203 4300,5772,7700,10212,13472,17673,23076,29988,38808,50008,64184,82070,
%U A261203 104560,132760,167996,211920,266512,334202,417902,521152,648224,804254,995432
%N A261203 Expansion of f(-x^6)^2 / (phi(-x) * phi(-x^9)) in powers of x where phi(), f() are Ramanujan theta functions.
%C A261203 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261203 G. C. Greubel, <a href="/A261203/b261203.txt">Table of n, a(n) for n = 0..2500</a>
%H A261203 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261203 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261203 Expansion of q^(-1/2) * eta(q^2) * eta(q^6)^2 * eta(q^18) / (eta(q)^2 * eta(q^9)^2) in powers of q.
%F A261203 Euler transform of period 18 sequence [ 2, 1, 2, 1, 2, -1, 2, 1, 4, 1, 2, -1, 2, 1, 2, 1, 2, 0, ...].
%F A261203 a(n) = A261154(2*n + 1).
%F A261203 Convolution inverse of A261202.
%F A261203 a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A261203 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 60*x^7 + ...
%e A261203 G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 14*q^9 + 24*q^11 + 38*q^13 + ...
%t A261203 a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 / (EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^9]), {x, 0, n}];
%o A261203 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^9 + A)^2), n))};
%Y A261203 Cf. A261154, A261202.
%K A261203 nonn
%O A261203 0,2
%A A261203 _Michael Somos_, Aug 11 2015