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A360875 Array read by antidiagonals: T(m,n) is the number of connected dominating sets in the rook graph K_m X K_n.

%I A360875 #10 Feb 16 2025 08:34:04
%S A360875 1,3,3,7,9,7,15,39,39,15,31,177,325,177,31,63,783,2931,2931,783,63,
%T A360875 127,3369,26077,51465,26077,3369,127,255,14199,225459,894675,894675,
%U A360875 225459,14199,255,511,58977,1901725,15195897,30331861,15195897,1901725,58977,511
%N A360875 Array read by antidiagonals: T(m,n) is the number of connected dominating sets in the rook graph K_m X K_n.
%H A360875 Andrew Howroyd, <a href="/A360875/b360875.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals).
%H A360875 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>
%H A360875 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%F A360875 T(m,n) = (Sum_{i=1..m} binomial(m,i) * A262307(n,i)) + (Sum_{j=1..n} binomial(n,j) * A262307(m,j)) - A262307(m,n).
%F A360875 T(m,n) = T(n,m).
%e A360875 Array begins:
%e A360875 =======================================================
%e A360875 m\n|  1    2      3        4          5           6 ...
%e A360875 ---+---------------------------------------------------
%e A360875 1  |  1    3      7       15         31          63 ...
%e A360875 2  |  3    9     39      177        783        3369 ...
%e A360875 3  |  7   39    325     2931      26077      225459 ...
%e A360875 4  | 15  177   2931    51465     894675    15195897 ...
%e A360875 5  | 31  783  26077   894675   30331861  1010163363 ...
%e A360875 6  | 63 3369 225459 15195897 1010163363 66273667449 ...
%e A360875   ...
%o A360875 (PARI) \\ S is A183109, T is A262307, U is this sequence.
%o A360875 G(M,N=M)={S=matrix(M, N); T=matrix(M, N); U=matrix(M, N);
%o A360875 for(m=1, M, for(n=1, N,
%o A360875   S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
%o A360875   T[m, n]=S[m, n]-sum(i=1, m-1, sum(j=1, n-1, T[i, j]*S[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j)));
%o A360875   U[m, n]=sum(i=1, m, binomial(m, i)*T[i, n])+sum(j=1, n, binomial(n, j)*T[m, j])-T[m, n] )); U
%o A360875 }
%o A360875 { my(A=G(7)); for(n=1, #A~, print(A[n,])) }
%Y A360875 Main diagonal is A289196.
%Y A360875 Rows 1..2 are A000225, A360876.
%Y A360875 Cf. A287274, A360873.
%K A360875 nonn,tabl
%O A360875 1,2
%A A360875 _Andrew Howroyd_, Feb 24 2023