A317278 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^n*n!/k!.

1, 1, 2, -15, -164, 4245, 46386, -4901939, 39141656, 11707820361, -671114863610, -29398709945319, 7385525824325364, -307076643365636963, -73748845974115224262, 14299745046516639280005, -237996466462017367478864, -377740669670216316717155055, 75515477307532501838072029326

Offset: 0

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

• sign

Comments

a(n) is the n-th term of the inverse 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

Crossrefs

Cf. A111884, A256467, A317277, A317279.

Programs

Magma

[1]cat[(&+[(-1)^(n+j)*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

Maple

A317278:= n-> `if`(n=0,1, add((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n));
seq(A317278(n), n=0..30); # G. C. Greubel, Mar 09 2021

Mathematica

Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] k^n n!/k!, {k, n}], {n, 18}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 + x))^k/k!, {k, n}], {x, 0, n}], {n, 18}]]

PARI

a(n) = if (n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 13 2022

Sage

[1]+[sum((-1)^(n+j)*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

Formula

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