A303208 Number of total dominating sets in the n X n rook graph.

0, 9, 334, 53731, 30844786, 66544564805, 556588617042914, 18376877842518517955, 2414913046805958120844234, 1267171440764716263069641387581, 2658150749788131925244338204731596650, 22299981643440069703358952237798936248817875

Offset: 1

Author

Eric W. Weisstein, Apr 19 2018

Keywords

• nonn

Links

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

Crossrefs

Main diagonal of A384116.

Cf. A287065, A289196, A303211, A347921.

Programs

Mathematica

b[0] = 1; b[n_] := (2^n - 1)^n + Sum[Binomial[n, i] Sum[(-1)^j (-1 + 2^(n - j))^i Binomial[n, j], {j, 0, n}], {i, n - 1}]; Table[Sum[(-1)^k Binomial[n, k]^2 k! b[n - k], {k, 0, n}], {n, 10}]

PARI

\\ here c(n) is A287065.
b(m, n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
c(n)=(2^n-1)^n + sum(i=1, n-1, b(n, i)*binomial(n, i));
a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)^2*k!*c(n-k))} \\ Andrew Howroyd, Apr 20 2018

Formula

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^2*k!*A287065(n-k). - Andrew Howroyd, Apr 20 2018

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Apr 20 2018

Extensions

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