A255050 G.f.: Product_{j>=1} 1/(1-x^j)^binomial(j+3,3).

1, 4, 20, 80, 305, 1072, 3622, 11676, 36450, 110240, 324936, 935076, 2635338, 7285560, 19795370, 52930360, 139462956, 362471020, 930186694, 2358867240, 5915606398, 14680528648, 36073675792, 87816701332, 211891552280, 506981067168, 1203337174120, 2834401172172

Offset

0,2

Links

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

Vaclav Kotesovec, Graph - The asymptotic ratio

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^C(j+3,3).

a(n) ~ Zeta(5)^(829/3600) * exp(11/72 - Zeta(3)/(4*Pi^2) + Zeta'(-3)/6 - 121*Zeta(3)^2 / (360*Zeta(5)) - Pi^6/(1800*Zeta(5)) + 11*Pi^8*Zeta(3) / (108000*Zeta(5)^2) - Pi^16/(194400000*Zeta(5)^3) + Pi^2 * n^(1/5)/ (6*2^(2/5) * Zeta(5)^(1/5)) - 11*Pi^4 * Zeta(3) * n^(1/5) / (900*2^(2/5)*Zeta(5)^(6/5)) + Pi^12 * n^(1/5) / (1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Zeta(3) * n^(2/5) / (6*2^(4/5) * Zeta(5)^(2/5)) - Pi^8 * n^(2/5) / (9000 * 2^(4/5) * Zeta(5)^(7/5)) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * Zeta(5)^(3/5)) + 5 * Zeta(5)^(1/5) * n^(4/5) / 2^(8/5)) / (A^(11/6) * 2^(971/1800) * 5^(1/2) * Pi * n^(2629/3600)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .

Maple

with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*binomial(d+3, 3), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50); # after Alois P. Heinz

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+3, 3], {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Column k=4 of A075196.

Cf. A005380, A217093, A255052.

Sequence in context: A258627 A082138 A074358 * A371408 A292540 A320934

Adjacent sequences: A255047 A255048 A255049 * A255051 A255052 A255053

Keyword

nonn

Author

Vaclav Kotesovec, Mar 08 2015

Status

approved