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
Examples
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 ... ...
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.
Programs
-
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, ])) }
Formula
T(n,m) = T(m,n).