A257655 Expansion of f(x^3, x^9) * f(x^6, x^6) / f(-x, -x^2) in powers of x where f(,) is Ramanujan's general theta function.

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 138, 188, 256, 348, 463, 614, 812, 1060, 1378, 1785, 2292, 2932, 3740, 4736, 5978, 7522, 9416, 11750, 14620, 18116, 22384, 27585, 33878, 41500, 50714, 61794, 75120, 91118, 110247, 133110, 160390, 192836, 231400, 277162

Offset: 0

Author

Michael Somos, Jul 25 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 22*x^7 + 33*x^8 + ...
G.f. = q + q^4 + 2*q^7 + 4*q^10 + 6*q^13 + 9*q^16 + 16*q^19 + 22*q^22 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A097196.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8) QPochhammer[ x]), {x, 0, n}];
eta[q_] := q^(1/24)*QPochhammer[q]; With[{nmax = 50}, CoefficientList[ Series[q^(-1/3)*eta[q^12]^5/(eta[q]*eta[q^3]*eta[q^24]^2), {x, 0, nmax}], x]] (* G. C. Greubel, Aug 02 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^12 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^24 + A)^2), n))};

Formula

Expansion of f(-x, -x^5) * f(x^6, x^6) / f(-x, -x) in powers of x where f(,) is Ramanujan's general theta function.

Expansion of q^(-1/3) * eta(q^12)^5 / (eta(q) * eta(q^3) * eta(q^24)^2) in powers of q.

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

a(n) = A097196(2*n).