A289196 Number of connected dominating sets in the n X n rook graph.

1, 9, 325, 51465, 30331861, 66273667449, 556170787050565, 18374555799096912585, 2414861959450912233421141, 1267166974391002542218440851129, 2658149210218078451926703769353958085, 22299979556058598891936157095746389850916425

Offset: 1

Author

Eric W. Weisstein, Jun 28 2017

Keywords

• nonn

Comments

A set of vertices in the n X n rook graph can be represented as a n X n binary matrix. The vertex set will be dominating if either every row contains a 1 or every column contains a 1. - Andrew Howroyd, Jul 18 2017

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Connected Dominating Set
• Eric Weisstein's World of Mathematics, Rook Graph

Crossrefs

Main diagonal of A360875.

Cf. A183109, A262307, A286189, A287065.

Programs

Mathematica

(* b = A183109, T = A262307 *) b[m_, n_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}]; T[, 1] = T[1, ] = 1; T[m_, n_] := T[m, n] = b[m, n] - Sum[T[i, j]*b[m-i, n-j]*Binomial[m-1, i-1]*Binomial[n, j], {i, 1, m-1}, {j, 1, n-1}]; a[n_] := T[n, n] + 2*Sum[ Binomial[n, k]*T[n, k], {k, 1, n-1}]; Array[a, 12] (* Jean-François Alcover, Oct 02 2017, after Andrew Howroyd *)

PARI

G(N)={S=matrix(N, N); T=matrix(N, N); U=matrix(N, N);
\\ S is A183109, T is A262307, U is m X n variant of this sequence.
for(m=1, N, for(n=1, N,
S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m, n]=S[m, n]-sum(i=1, m-1, sum(j=1, n-1, T[i, j]*S[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j)));
U[m, n]=sum(i=1, m, binomial(m, i)*T[i, n])+sum(j=1, n, binomial(n,j)*T[m, j])-T[m,n] )); U}
a(n)=G(n)[n, n]; \\ Andrew Howroyd, Jul 18 2017

Formula

a(n) = A262307(n,n) + 2*Sum_{k=1..n-1} binomial(n,k) * A262307(n,k). - Andrew Howroyd, Jul 18 2017

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jul 18 2017