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

0,2

Links

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

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A187427.

Sequence in context: A126943 A209399 A192974 * A316878 A036368 A006071

Adjacent sequences: A187425 A187426 A187427 * A187429 A187430 A187431

Keyword

nonn

Author

Michael Somos, Mar 09 2011

Status

approved