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A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.

%I A261445 #15 Feb 16 2025 08:33:26
%S A261445 1,4,9,14,16,18,21,28,36,38,40,36,43,52,54,62,56,72,74,72,81,64,88,90,
%T A261445 98,100,72,110,112,126,133,104,126,108,136,144,112,148,144,158,144,
%U A261445 144,183,172,180,182,152,162,194,196,198,160,216,216,180,224,189,230
%N A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.
%C A261445 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261445 G. C. Greubel, <a href="/A261445/b261445.txt">Table of n, a(n) for n = 0..1000</a>
%H A261445 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261445 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261445 Expansion of f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
%F A261445 Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.
%F A261445 Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].
%F A261445 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - _Michael Somos_, Nov 13 2015
%F A261445 a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - _Michael Somos_, Nov 13 2015
%F A261445 a(3*n + 1) = 4 * A260165(n). a(3*n + 2) = 9 * A263021(n). - _Michael Somos_, Nov 13 2015
%e A261445 G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ...
%e A261445 G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...
%t A261445 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
%t A261445 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Nov 13 2015 *)
%t A261445 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* _Michael Somos_, Nov 13 2015 *)
%o A261445 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};
%Y A261445 Cf. A124815, A209613, A260109 A260165, A260301, A263021.
%K A261445 nonn
%O A261445 0,2
%A A261445 _Michael Somos_, Aug 18 2015