A341815 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k^n.

1, 1, 12, 270, 8960, 406250, 23293872, 1617774592, 132075970560, 12397121784954, 1315528361642000, 155743010418063860, 20351866171905066240, 2909818652684404979440, 451849287590990124662400, 75730203998219999637000000, 13625593688459657260608782336, 2619521322904712777031960349850

Offset

0,3

Comments

For m>0, Sum_{k=0..n} binomial(n,k)^m * k^n ~ c(m) * d^n * n! / n^(m/2), where d = (1 + m*LambertW(exp(-1/m)/m))^(m-1) / (m^m * LambertW(exp(-1/m)/m)^m) and c(m) is a constant independent of n.

Links

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

Formula

a(n) ~ c * d^n * n! / n^(3/2), where d = (1 + 3*LambertW(exp(-1/3)/3))^2 / (27 * LambertW(exp(-1/3)/3)^3) = 12.3645613141726293982008517178673172577947617775... and c = 0.143687082995832067469009730530027989920523409582173778129054767279...

Mathematica

Join[{1}, Table[Sum[k^n * Binomial[n, k]^3, {k, 0, n}], {n, 1, 20}]]

Crossrefs

Cf. A072034, A336828.

Sequence in context: A239779 A107485 A293270 * A117415 A152129 A115461

Adjacent sequences: A341812 A341813 A341814 * A341816 A341817 A341818

Keyword

nonn

Author

Vaclav Kotesovec, Feb 20 2021

Status

approved