A303211 Number of minimum total dominating sets in the n X n rook graph.

0, 4, 6, 80, 410, 5112, 48818, 695424, 9589266, 162327800, 2869193162, 57451559904, 1225220612954, 28560612445848, 709917843398850, 18943086191785472, 536695850359985186, 16151064034012994808, 513345798896635886906, 17206881800061632191200

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Rook Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Formula

a(n) = Sum_{k=0,..n} (-1)^k*binomial(n,k)^2*k!*(2*(n-k)^(n-k) - (n-k)!). - Andrew Howroyd, Apr 20 2018

a(n) ~ 2 * (exp(1) - 1)^(n + 1/2) * n^n / exp(n + 1/2). - Vaclav Kotesovec, Apr 20 2018

Mathematica

Table[(-1)^n n! + Sum[(-1)^k Binomial[n, k]^2 k! (2 (n - k)^(n - k) - (n - k)!), {k, 0, n - 1}], {n, 20}]
2 Table[(-1)^n n! + Sum[(-1)^k Binomial[n, k]^2 k! (n - k)^(n - k), {k, 0, n - 1}], {n, 20}] (* Eric W. Weisstein, Jan 18 2019 *)

Prog

(PARI) a(n) = {sum(k=0, n, (-1)^k*binomial(n, k)^2*k!*(2*(n-k)^(n-k) - (n-k)!))} \\ Andrew Howroyd, Apr 20 2018

Crossrefs

Cf. A248744, A303208.

Sequence in context: A197315 A061216 A058163 * A264374 A013127 A012970

Adjacent sequences: A303208 A303209 A303210 * A303212 A303213 A303214

Keyword

nonn

Author

Eric W. Weisstein, Apr 19 2018

Extensions

Terms a(6) and beyond from Andrew Howroyd, Apr 20 2018

Status

approved