A344052 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*E1(n, k).

1, -1, -1, 8, 19, -276, -1002, 21216, 103395, -2881180, -17620142, 609297072, 4483215086, -185182296040, -1592692090420, 76512069014400, 753146574607395, -41256108712556460, -457383584443526790, 28138583115102810000, 346933879489006727610, -23683708768534714984920

Offset: 0

Author

Peter Luschny, May 10 2021

Keywords

• sign

Comments

Inverse binomial convolution of the first order Eulerian numbers (A173018).

Crossrefs

Cf. A173018, A011818 (binomial convolution).

Programs

Maple

A344052 := n -> add((-1)^(n-k)*binomial(n, k)*combinat:-eulerian1(n, k), k=0..n):
seq(A344052(n), n=0..21);

Mathematica

a[n_] := Sum[Sum[(-1)^(n - k - j)(j - k - 1)^n Binomial[n, k] Binomial[n + 1, j], {j, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(n - k - j)*(j - k - 1)^n * binomial(n, k)* binomial(n+1, j).