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A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function. 3

%I #11 Mar 12 2021 22:24:48

%S 1,2,2,4,5,6,6,4,7,4,6,8,4,10,8,12,8,6,14,8,11,6,8,8,8,14,6,12,15,14,

%T 14,8,12,14,12,16,8,10,14,16,16,12,12,12,16,10,10,8,19,20,20,8,12,24,

%U 14,24,12,16,14,16,21,10,14,28,16,12,14,12,16,16,30,12

%N Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A263433/b263433.txt">Table of n, a(n) for n = 0..2500</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^6)^4 / (eta(q)^2 * eta(q^12)^2) in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263444.

%F a(n) = A261426(2*n) = A045832(6*n). 3 * a(n) = A005889(6*n).

%e G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 4*x^7 + 7*x^8 + ...

%e G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 6*q^37 + 4*q^43 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 4, 0, x], {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x + A)^2 * eta(x^12 + A)^2), n))};

%Y Cf. A005889, A045832, A261426, A263444.

%K nonn

%O 0,2

%A _Michael Somos_, Oct 18 2015

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