A249781 Expansion of q * f(-q)^2 * f(-q^15)^2 * chi(-q) * chi(-q^15) / (chi(-q^3) * chi(-q^5)) in powers of q where f(), chi() are Ramanujan theta functions.

1, -3, 1, 3, -1, 1, -4, 1, 1, -1, 0, -1, 2, 4, -1, -5, 6, -3, -4, 1, -4, 8, 0, -3, 1, 2, 1, -4, -6, 3, 8, 1, 0, -10, 4, 3, 2, -4, 2, 3, -6, 4, -4, -8, -1, 0, 0, 7, 9, -3, 6, -2, -6, 1, 0, 4, -4, 10, 0, -3, -10, -8, -4, 11, -2, -8, -4, 10, 0, -4, 0, 1, 2, 18

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^3 * eta(q^6) * eta(q^10) * eta(q^15)^3 / (eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30)) in powers of q.

Euler transform of period 30 sequence [ -3, -2, -2, -2, -2, -2, -3, -2, -2, -2, -3, -2, -3, -2, -4, -2, -3, -2, -3, -2, -2, -2, -3, -2, -2, -2, -2, -2, -3, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 60 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A249371.

Example

G.f. = q - 3*q^2 + q^3 + 3*q^4 - q^5 + q^6 - 4*q^7 + q^8 + q^9 - q^10 - q^12 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^3* eta[q^6]*eta[q^10]*eta[q^15]^3/(eta[q^2]*eta[q^3]*eta[q^5]*eta[q^30]), {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 15 2018 *)

Prog

(PARI) {a(n) = my(A); n-=1; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A)^3 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^3*eta(q^6)*eta(q^10)*eta(q^15)^3/(eta(q^2)*eta(q^3)*eta(q^5)*eta(q^30))) \\ Altug Alkan, Mar 16 2018
(Magma) A := Basis( CuspForms( Gamma0(30), 2), 75); A[1] - 3*A[2] + A[3];

Crossrefs

Cf. A249371.

Sequence in context: A225212 A091088 A335915 * A342041 A115627 A128218

Adjacent sequences: A249778 A249779 A249780 * A249782 A249783 A249784

Keyword

sign

Author

Michael Somos, Nov 05 2014

Status

approved