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
Keywords
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
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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