A132609 Antidiagonal sum of table A072590(n,k) = n^(k-1)*k^(n-1) for n>=1.

1, 2, 6, 26, 147, 1026, 8532, 82394, 906485, 11194402, 153347766, 2307805402, 37851581159, 672037936898, 12841521329896, 262772642843802, 5733086299727913, 132853067341477538, 3258726189638877610

Offset: 1

Author

Paul D. Hanna, Aug 26 2007

Keywords

• nonn

Comments

A072590(n,k) equals the number of spanning trees in complete bipartite graph K(n,k).

Also the number of minimum connected dominating sets of the (n+1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Jun 03 2024 and Mar 05 2025

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..429
• Eric Weisstein's World of Mathematics, Connected Dominating Set.
• Eric Weisstein's World of Mathematics, Triangular Honeycomb Bishop Graph.

Crossrefs

Cf. A072590, A062817; A132608.

Programs

Mathematica

Table[Sum[(n - k + 1)^(k - 1) k^(n - k), {k, n}], {n, 30}] (* Harvey P. Dale, Jun 26 2021 *)

PARI

a(n)=sum(k=1,n,(n-k+1)^(k-1)*k^(n-k))

Formula

a(n) = Sum_{k=1..n} (n-k+1)^(k-1)*k^(n-k) for n>=1.

a(n) ~ sqrt(2*Pi/3) * exp(1) * n^(n - 1/2) / 2^n. - Vaclav Kotesovec, Nov 22 2021