A133739 Expansion of q * (psi(q^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 9, 31, 45, 6, -45, 8, 117, 121, 54, 12, -9, 14, 72, 186, 261, 18, -207, 20, 270, 248, 108, 24, 63, 31, 126, 391, 360, 30, -270, 32, 549, 372, 162, 48, -171, 38, 180, 434, 702, 42, -360, 44, 540, 726, 216, 48, 207, 57, 279, 558, 630, 54, -693, 72, 936, 620

Offset

1,2

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 = 1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^23 * eta(q^3)^3 * eta(q^12)^6 / (eta(q)^9 * eta(q^4)^10 * eta(q^6)^9) in powers of q.

Euler transform of period 12 sequence [ 9, -14, 6, -4, 9, -8, 9, -4, 6, -14, 9, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134078.

G.f.: f(x) + 6 * f(x^2) + 27 * f(x^3) + 20 * f(x^4) - 162 * f(x^6) + 108 * f(x^12) where f() is the g.f. of A000203.

a(4*n + 2) = 9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

Example

G.f. = q + 9*q^2 + 31*q^3 + 45*q^4 + 6*q^5 - 45*q^6 + 8*q^7 + 117*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(3/2)])^3 (EllipticTheta[ 3, 0, x]^5 / EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
QP=QPochhammer; CoefficientList[Series[QP[q^2]^23*QP[q^3]^3*QP[q^12]^6/( QP[q]^9*QP[q^4]^10*QP[q^6]^9), {q, 0, 50}], q] (* G. C. Greubel, Nov 16 2018 *)

Prog

(PARI) {a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^23 * eta(x^3 + A)^3 * eta(x^12 + A)^6 / (eta(x + A)^9 * eta(x^4 + A)^10 * eta(x^6 + A)^9), n))};
(Magma) A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* Michael Somos, Oct 30 2015 */

Crossrefs

Cf. A000203, A098098, A134077, A134078.

Sequence in context: A054310 A072887 A329651 * A266397 A288419 A168297

Adjacent sequences: A133736 A133737 A133738 * A133740 A133741 A133742

Keyword

sign

Author

Michael Somos, Oct 06 2007

Status

approved