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A262709 Expansion of f(-x^4, -x^4) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function. 1

%I #9 Mar 12 2021 22:24:48

%S 1,-1,0,0,-2,1,0,0,1,2,0,0,-2,0,0,0,3,-2,0,0,-2,-3,0,0,2,2,0,0,0,0,0,

%T 0,2,-1,0,0,-2,2,0,0,1,2,0,0,-4,0,0,0,0,-2,0,0,-2,0,0,0,3,2,0,0,-2,0,

%U 0,0,2,-3,0,0,0,2,0,0,4,0,0,0,-2,0,0,0,2,-2

%N Expansion of f(-x^4, -x^4) * f(-x, -x^5) 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="/A262709/b262709.txt">Table of n, a(n) for n = 0..1000</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 q^(-1/3) * eta(q) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^8)) in powers of q.

%F Euler transform of period 24 sequence [ -1, 0, 0, -2, -1, -1, -1, -1, 0, 0, -1, -3, -1, 0, 0, -1, -1, -1, -1, -2, 0, 0, -1, -2, ...].

%e G.f. = 1 - x - 2*x^4 + x^5 + x^8 + 2*x^9 - 2*x^12 + 3*x^16 - 2*x^17 + ...

%e G.f. = q - q^4 - 2*q^13 + q^16 + q^25 + 2*q^28 - 2*q^37 + 3*q^49 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A)), n))};

%K sign

%O 0,5

%A _Michael Somos_, Sep 28 2015

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)