A333043 G.f.: exp(Sum_{k>=1} (5*k)!/k!^5 * x^k/k).

1, 120, 63900, 63148000, 85136103750, 137629764435024, 250331826090382280, 494436455370401985600, 1037731227148399567352625, 2281874234819846601146115000, 5205960892339635531670022801628, 12237148815599682784939438806708960, 29483782935554473122496294160376815950

Offset: 0

Author

Vaclav Kotesovec, Mar 06 2020

Keywords

• nonn

Comments

In general, if r>=2, m>0 and g.f. = exp(m * Sum_{k>=1} (r*k)!/k!^r * x^k/k), then a(n) ~ c(r,m) * m * r^(r*n + 1/2) / ((2*Pi)^((r-1)/2) * n^((r+1)/2)) , where c(r,m) = exp((m * r! / r^r) * HypergeometricPFQ[{1, 1, (r+1)/r, (r+2)/r, ... , (2*r-1)/r}, {2, 2, ...r-times... 2, 2}, 1]). - Vaclav Kotesovec, Feb 16 2024

Crossrefs

Cf. A000108, A008978, A229451, A333042, A370295.

Programs

Mathematica

CoefficientList[Series[Exp[Sum[(5*k)!/k!^5*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
CoefficientList[Series[Exp[120*x*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 3125*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2024 *)

Formula

a(n) ~ c * 5^(5*n)/n^3, where c = sqrt(5) * exp(24*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 1] / 625) / (4*Pi^2) = 0.05943406... - Vaclav Kotesovec, Mar 06 2020, updated Feb 16 2024

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A008978(k) * a(n-k). - Seiichi Manyama, Feb 09 2024