A285215 Expansion of Product_{k>=1} (1 - x^(4*k))^(4*k) / (1 - x^k)^k.

1, 1, 3, 6, 9, 20, 36, 62, 106, 184, 302, 503, 829, 1325, 2119, 3367, 5282, 8227, 12740, 19550, 29849, 45300, 68325, 102495, 152998, 227249, 336005, 494597, 724875, 1058213, 1538860, 2229370, 3218304, 4630015, 6638728, 9488894, 13520995, 19208916, 27211430

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>=0} 1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)).

a(n) ~ exp(-1/4 + 2^(-4/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(5/4) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

Mathematica

nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))^(4*k)/((1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

Prog

(PARI) x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(4*k + 1))^(4*k + 1)*(1 - x^(4*k + 2))^(4*k + 2)*(1 - x^(4*k + 3))^(4*k + 3)))) \\ Indranil Ghosh, Apr 15 2017

Crossrefs

Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), this sequence (m=4), A285246 (m=5).

Cf. A285262, A285284.

Sequence in context: A018186 A223504 A322949 * A015938 A116614 A089001

Adjacent sequences: A285212 A285213 A285214 * A285216 A285217 A285218

Keyword

nonn

Author

Seiichi Manyama, Apr 15 2017

Status

approved