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

0,3

Comments

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

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

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).

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A097196.

Sequence in context: A372542 A226007 A372632 * A318026 A173241 A096398

Adjacent sequences: A257652 A257653 A257654 * A257656 A257657 A257658

Keyword

nonn

Author

Michael Somos, Jul 25 2015

Status

approved