A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.

1, -3, 0, 9, -12, 0, 27, -42, 0, 82, -111, 0, 207, -279, 0, 486, -630, 0, 1055, -1362, 0, 2205, -2775, 0, 4374, -5472, 0, 8427, -10389, 0, 15696, -19224, 0, 28539, -34614, 0, 50630, -61059, 0, 88119, -105483, 0, 150417, -179178, 0, 252727, -299325, 0, 418068

Offset: 0

Author

Michael Somos, Mar 09 2011

Keywords

• sign

Examples

			1 - 3*x + 9*x^3 - 12*x^4 + 27*x^6 - 42*x^7 + 82*x^9 - 111*x^10 + ...
q^-3 - 3*q^5 + 9*q^21 - 12*q^29 + 27*q^45 - 42*q^53 + 82*q^69 - 111*q^77 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500

Crossrefs

Cf. A053762, A187248.

Programs

Mathematica

QP = QPochhammer; s = QP[q]^3/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(3/8) *eta[q]^3/ eta[q^3]^4, {q, 0, 50}], q] (* G. C. Greubel, Aug 14 2018 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / eta(x^3 + A)^4, n))}

Formula

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = (9/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187428.

G.f.: Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k))^4.

a(3*n) = A053762(n). a(3*n + 1) = -3 * A187428(n). a(3*n + 2) = 0.