A368831 -id:A368831 - cp's OEIS frontend

A248744 Number of different ways one can attack all squares on an n X n chessboard with n rooks.

1, 1, 6, 48, 488, 6130, 92592, 1642046, 33514112, 774478098, 19996371200, 570583424422, 17831721894912, 605743986163706, 22223926472824832, 875786473087350750, 36893467224629215232, 1654480168085245432354, 78692809748219369422848, 3956839189675526769415958

Offset: 0

Stephen Penrice, Apr 09 2017

nonn

Number of minimum (and minimal) dominating sets in the n X n rook graph. - Eric W. Weisstein, Jun 20 2017 and Aug 02 2017

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Vol. 1: Combinatorial Analysis and Probability Theory, Dover Publications, 1987, p. 77

Eric Weisstein's World of Mathematics, Minimal Dominating Set
Eric Weisstein's World of Mathematics, Minimum Dominating Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal of A290632 and of A368831.

Cf. A000142, A000312.

A248744:=n->2*n^n-n!: seq(A248744(n), n=0..25); # Wesley Ivan Hurt, Nov 30 2017

Mathematica
```
Table[2 n^n - n!, {n, 20}]
```

a(n) = 2*n^n - n!.

A287065 Number of dominating sets on the n X n rook graph.

Original entry on oeis.org

1, 11, 421, 59747, 32260381, 67680006971, 559876911043381, 18412604442711949187, 2416403019417984915336061, 1267413006543912045144741284411, 2658304092145691708492995820522716981, 22300364428188338185156192161829091442585827

Offset: 1

Eric W. Weisstein, May 19 2017

nonn

Number of {0,1} n X n matrices with no zero rows or no zero columns. - Geoffrey Critzer, Jan 15 2024

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

Main diagonal of A287274.
Row sums of A368831.
Cf. A183109, A002720, A048291, A173403.

Table[(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}], {n, 20}] (* Eric W. Weisstein, May 27 2017 *)

b(m,n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
a(n)=(2^n-1)^n + sum(i=1,n-1,b(n,i)*binomial(n,i)); \\ Andrew Howroyd, May 22 2017

a(n) = (2^n-1)^n + Sum_{i=1..n-1} binomial(n,i) * A183109(n,i). - Andrew Howroyd, May 22 2017

a(6)-a(12) from Andrew Howroyd, May 22 2017

cp's OEIS Frontend

A248744 Number of different ways one can attack all squares on an n X n chessboard with n rooks.

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