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.

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

0,12

Links

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.

Formula

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

Example

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 ...
  ...

Prog

(PARI)
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, ])) }

Crossrefs

Main diagonal is A347921.

Cf. A290632, A384116, A384117.

Sequence in context: A306234 A290057 A384117 * A249790 A302713 A136206

Adjacent sequences: A384115 A384116 A384117 * A384119 A384120 A384121

Keyword

nonn,tabl

Author

Andrew Howroyd, May 19 2025

Status

approved