A354943 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).

1, 2, 22, 438, 12824, 496370, 23914512, 1379269094, 92667551104, 7108231236066, 612974464428800, 58702772664490262, 6181602019316333568, 709911177607125141362, 88301595129435811723264, 11825985945777638231211750, 1696696168760520436580974592, 259624546758869333450285984066

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.

a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - Vaclav Kotesovec, Jul 01 2022, updated Mar 17 2024

Mathematica

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)^3 * k! * n^(n-k)); \\ Michel Marcus, Jun 12 2022

Crossrefs

Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.

Sequence in context: A328158 A266522 A360304 * A084949 A276454 A137076

Adjacent sequences: A354940 A354941 A354942 * A354944 A354945 A354946

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 12 2022

Status

approved