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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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 wehre 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: A179997 A101756 A226007 * A318026 A173241 A096398

Adjacent sequences:  A257652 A257653 A257654 * A257656 A257657 A257658

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 25 2015

STATUS

approved

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Last modified November 15 11:35 EST 2018. Contains 317238 sequences. (Running on oeis4.)