A384117 -id:A384117 - cp's OEIS frontend

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

Eric W. Weisstein, Apr 19 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal of A384117.

Cf. A248744, A303208, A347921.

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 *)

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) = 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

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

A384116 Array read by antidiagonals: T(n,m) is the number of total dominating sets in the n X m rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 4, 9, 4, 1, 1, 11, 39, 39, 11, 1, 1, 26, 183, 334, 183, 26, 1, 1, 57, 833, 3087, 3087, 833, 57, 1, 1, 120, 3629, 27472, 53731, 27472, 3629, 120, 1, 1, 247, 15291, 236127, 922515, 922515, 236127, 15291, 247, 1, 1, 502, 63051, 1975246, 15524639, 30844786, 15524639, 1975246, 63051, 502, 1

Offset: 0

Andrew Howroyd, May 19 2025

			Array begins:
=================================================================
n\m | 0   1     2       3         4           5             6 ...
----+------------------------------------------------------------
  0 | 1   1     1       1         1           1             1 ...
  1 | 1   0     1       4        11          26            57 ...
  2 | 1   1     9      39       183         833          3629 ...
  3 | 1   4    39     334      3087       27472        236127 ...
  4 | 1  11   183    3087     53731      922515      15524639 ...
  5 | 1  26   833   27472    922515    30844786    1019569593 ...
  6 | 1  57  3629  236127  15524639  1019569593   66544564805 ...
  7 | 1 120 15291 1975246 256594143 33329148492 4314985562475 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Total Dominating Set.

Main diagonal is A303208.
Column 0 is A000012.
Column 1 is A000295(n), n > 0.
Column 2 is A287063(n), n > 1.
Cf. A287274, A384117, A384118.

B(n,m) = {sum(i=0, min(n,m), (-1)^i*binomial(n,i)*binomial(m,i)*i!*(2^(n-i)-1)^(m-i))}
T(n,m) = {B(n,m) - sum(i=1, m, (-1)^i*binomial(m,i)*B(m-i,n))}

T(n,m) = B(n,m) - Sum_{i=1..m} (-1)^i*binomial(m,i)*B(m-i,n), where B(n,m) = Sum_{i=0..m} (-1)^i*binomial(n,i)*binomial(m,i)*i!*(2^(n-i)-1)^(m-i).

T(n,m) = T(m,n).

A384118 Array read by antidiagonals: T(n,m) is the number of minimal total dominating sets in the n X m rook graph K_n X K_m.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 6, 5, 5, 6, 1, 1, 10, 12, 51, 12, 10, 1, 1, 15, 37, 97, 97, 37, 15, 1, 1, 21, 98, 218, 368, 218, 98, 21, 1, 1, 28, 219, 519, 2229, 2229, 519, 219, 28, 1, 1, 36, 430, 1417, 6232, 7310, 6232, 1417, 430, 36, 1

Offset: 0

Andrew Howroyd, May 19 2025

			Array begins:
=====================================================
n\m | 0  1   2    3     4      5       6        7 ...
----+------------------------------------------------
  0 | 1  1   1    1     1      1       1        1 ...
  1 | 1  0   1    3     6     10      15       21 ...
  2 | 1  1   4    5    12     37      98      219 ...
  3 | 1  3   5   51    97    218     519     1417 ...
  4 | 1  6  12   97   368   2229    6232    16013 ...
  5 | 1 10  37  218  2229   7310   44491   172387 ...
  6 | 1 15  98  519  6232  44491  301572  1345693 ...
  7 | 1 21 219 1417 16013 172387 1345693 10893008 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Rook Graph.

Main diagonal is A347921.

Cf. A290632, A384116, A384117.

B(n,m)={ my(M=matrix(n+1,m+1)); for(n=1, n, M[n+1,1]=1; for(m=1, m, M[n+1,m+1] = if(n>2, binomial(n,2)*M[n-1,m]) + sum(i=2, m, binomial(m-1,i-1)*(n*M[n, m-i+1] + if(i>=3&&i<=n, binomial(n,i-1)*i!*M[n-i+2,m-i+1] ) )))); M}
A(n,m)={ my(M=B(m,n) + B(n,m)~); M[1,1]=1; for(i=1, m, for(j=1, n, if((i+j)%3==0 && j<=2*i && i<=2*j, my(t=(i+j)/3); M[i+1,j+1] += binomial(i,j-t)*binomial(j,i-t)*(2*(j-t))!*(2*(i-t))!/2^t ))); M}
{ my(T=A(8,8)); for(i=1, #T, print(T[i, ])) }

T(n,m) = T(m,n).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A384116 Array read by antidiagonals: T(n,m) is the number of total dominating sets in the n X m rook graph K_n X K_m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A384118 Array read by antidiagonals: T(n,m) is the number of minimal total dominating sets in the n X m rook graph K_n X K_m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula