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A360850 Array read by antidiagonals: T(m,n) is the number of (undirected) paths in the complete bipartite graph K_{m,n}.

%I A360850 #13 Feb 16 2025 08:34:04
%S A360850 1,3,3,6,12,6,10,33,33,10,15,72,135,72,15,21,135,438,438,135,21,28,
%T A360850 228,1140,2224,1140,228,28,36,357,2511,8850,8850,2511,357,36,45,528,
%U A360850 4893,27480,55725,27480,4893,528,45,55,747,8700,70462,265665,265665,70462,8700,747,55
%N A360850 Array read by antidiagonals: T(m,n) is the number of (undirected) paths in the complete bipartite graph K_{m,n}.
%C A360850 T(m,n) is the number of induced paths including zero length paths in the m X n rook graph. This is also the number of induced trees in these graphs since these are the only induced trees.
%H A360850 Andrew Howroyd, <a href="/A360850/b360850.txt">Table of n, a(n) for n = 1..1275</a>
%H A360850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H A360850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>.
%H A360850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%F A360850 T(m,n) = Sum_{j=1..min(m,n)} j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j).
%F A360850 T(m,n) = T(n,m).
%e A360850 Array begins:
%e A360850 ===================================================
%e A360850 m\n|  1   2    3     4      5        6        7 ...
%e A360850 ---+-----------------------------------------------
%e A360850 1  |  1   3    6    10     15       21       28 ...
%e A360850 2  |  3  12   33    72    135      228      357 ...
%e A360850 3  |  6  33  135   438   1140     2511     4893 ...
%e A360850 4  | 10  72  438  2224   8850    27480    70462 ...
%e A360850 5  | 15 135 1140  8850  55725   265665   962010 ...
%e A360850 6  | 21 228 2511 27480 265665  2006316 11158203 ...
%e A360850 7  | 28 357 4893 70462 962010 11158203 98309827 ...
%e A360850    ...
%o A360850 (PARI) T(m,n) = sum(j=1, min(m,n), j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j))
%Y A360850 Main diagonal is A288035.
%Y A360850 Rows 1..2 are A000217, A054602.
%Y A360850 Cf. A360849 (cycles), A360851.
%K A360850 nonn,tabl
%O A360850 1,2
%A A360850 _Andrew Howroyd_, Feb 23 2023