A245668 - OEIS

%I #14 Sep 08 2022 08:46:09

%S 1,-3,3,-1,-3,6,-3,0,3,3,-12,6,-1,-12,12,0,-3,12,9,-12,6,-6,-12,0,-3,

%T -15,18,5,0,18,-6,0,3,-6,-24,12,3,-12,18,0,-12,24,-6,-12,6,18,-24,0,

%U -1,-27,21,-6,-12,18,15,0,12,-6,-12,18,0,-36,24,0,-3,24,-12

%N Expansion of (chi(q^3) * psi(-q))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A245668/b245668.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 phi(q^3) * psi(-q)^3 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.

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

%F Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.

%F a(3*n + 1) = -3 * A213056(n). a(6*n + 2) = 3 * A213592(n). a(6*n + 5) = 6 * A213607(n). a(8*n + 7) = 0.

%F Convolution cube of A089807.

%e G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q,0,n}];

%t a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q,0,n}];

%t a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q,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) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};

%o (Magma) A := Basis( ModularForms( Gamma0(12), 3/2), 67);  A[1] - 3*A[2] + 3*A[3];

%Y Cf. A089807, A213592, A213607, A245669.

%K sign

%O 0,2

%A _Michael Somos_, Jul 28 2014