A285262 -id:A285262

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A285215

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 15 2017

STATUS

approved

A285263

Expansion of Product_{k>=1} ((1-x^(5*k))/(1-x^k))^k.

+10
3

1, 1, 3, 6, 13, 23, 47, 83, 154, 269, 474, 809, 1387, 2313, 3859, 6330, 10341, 16680, 26790, 42586, 67375, 105731, 165097, 256052, 395248, 606501, 926502, 1408048, 2130788, 3209643, 4815595, 7194875, 10709843, 15881236, 23467805, 34556842, 50720003, 74200845

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

In general, if m > 1 and g.f. = Product_{k>=1} ((1-x^(m*k))/(1-x^k))^k, then a(n, m) ~ exp(3 * 2^(-2/3) * ((1-1/m^2)*Zeta(3))^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(1/3) * sqrt(3*Pi) * m^(1/12) * n^(2/3)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ exp(2^(1/3) * 3^(4/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/6) / (3^(1/3) * 5^(5/12) * sqrt(Pi) * n^(2/3)).

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[((1-x^(5*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A026007 (m=2), A263346 (m=3), A285262 (m=4).

Cf. A285246.