A317277 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*k^n*n!/k!; a(0) = 1.

1, 1, 6, 81, 1828, 60565, 2734926, 160109005, 11724156648, 1045312448841, 111114793839610, 13845807451708441, 1994597720747571468, 328351264019737949341, 61162428777982281583302, 12782305566531823350524805, 2975150384583838798131401296, 766253903501365584725344992529

Offset

0,3

Comments

a(n) is the n-th term of the Lah transform of the n-th powers.

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

N. J. A. Sloane, Transforms

Formula

a(n) = n! * [x^n] Sum_{k>=0} k^n*(x/(1 - x))^k/k!.

Maple

A317277:= n-> `if`(n=0, 1, add(binomial(n-1, j-1)*binomial(n, j)*(n-j)!*j^n, j=0..n)); seq(A317277(n), n=0..30); # G. C. Greubel, Mar 09 2021

Mathematica

Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^n n!/k!, {k, n}], {n, 17}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 - x))^k/k!, {k, n}], {x, 0, n}], {n, 17}]]

Prog

(Sage) [1]+[sum(binomial(n-1, j-1)*binomial(n, j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021
(Magma) [1]cat[(&+[Binomial(n-1, j-1)*Binomial(n, j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021
(PARI) a(n) = if (n==0, 1, sum(k=0, n, binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 15 2022

Crossrefs

Cf. A000262, A052852, A103194, A293145, A317278, A317279.

Sequence in context: A349651 A379940 A193265 * A138457 A252821 A371765

Adjacent sequences: A317274 A317275 A317276 * A317278 A317279 A317280

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 25 2018

Extensions

Name edited by Michel Marcus, Jun 15 2022

Status

approved