A370295 G.f.: exp(Sum_{k>=1} (5*k)!/(5!*k!^5) * x^k/k).

1, 1, 473, 467606, 637121154, 1039792179805, 1905441263652576, 3785382599457953517, 7981116324798212651066, 17613760342120835610374245, 40303877398793645855018120732, 94970269248783993542201925505548, 229287077006842005926064077532676555, 565001770629439341048001870559581136157

Offset

0,3

Comments

In general, for m>=2, if g.f. = exp(Sum_{k>=1} (m*k)!/(m!*k!^m) * x^k/k), then a(n,m) ~ c(m) * m^(m*n) / n^((m+1)/2), where c(m) = exp(HypergeometricPFQ[{1, 1, (m+1)/m, (m+2)/m, ... , (2*m-1)/m}, {2, 2, ...m-times... 2, 2}, 1] / m^m) / (m! * (2*Pi)^((m-1)/2) / sqrt(m)).

Limit_{m->oo} c(m) / (exp(m)/(m^m*(2*Pi)^(m/2))) = 1.

Links

Table of n, a(n) for n=0..13.

Formula

G.f. A(x) = G(x)^(1/120), where G(x) is the g.f. for A333043.

a(n) ~ c * 5^(5*n)/n^3, where c = exp(HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 1] / 3125) / (96*sqrt(5)*Pi^2) = 0.00047219161473962545263459216995582653262467228952818554361164671183728...

Mathematica

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

Crossrefs

Cf. A008978, A322252, A229452, A370294, A333043.

Sequence in context: A235897 A215738 A348806 * A045302 A187612 A237679

Adjacent sequences: A370292 A370293 A370294 * A370296 A370297 A370298

Keyword

nonn

Author

Vaclav Kotesovec, Feb 14 2024

Status

approved