A187428 Expansion of q^(-5/24) * eta(q^3)^3 / eta(q)^4 in powers of q.

1, 4, 14, 37, 93, 210, 454, 925, 1824, 3463, 6408, 11538, 20353, 35161, 59726, 99775, 164337, 266978, 428521, 679861, 1067415, 1659205, 2555617, 3902055, 5909867, 8881849, 13252334, 19637281, 28909989, 42297267, 61520450, 88976461, 127996994

Offset: 0

Author

Michael Somos, Mar 09 2011

Keywords

• nonn

Examples

			1 + 4*x + 14*x^2 + 37*x^3 + 93*x^4 + 210*x^5 + 454*x^6 + 925*x^7 + ...
q^5 + 4*q^29 + 14*q^53 + 37*q^77 + 93*q^101 + 210*q^125 + 454*q^149 + ...
		

Links

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

Crossrefs

Cf. A187427.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 - x^(3*k))^3 / (1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-5/24) *eta[q^3]^3/eta[q]^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^3 + A)^3 / eta(x + A)^4, n))}

Formula

Euler transform of period 3 sequence [ 4, 4, 1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 648^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187427.

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

a(n) ~ exp(sqrt(2*n)*Pi)/(12*sqrt(3)*n). - Vaclav Kotesovec, Sep 07 2015