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%I #8 Feb 27 2023 22:51:36
%S 0,1,1,3,8,3,6,27,27,6,10,64,126,64,10,15,125,426,426,125,15,21,216,
%T 1125,2208,1125,216,21,28,343,2493,8830,8830,2493,343,28,36,512,4872,
%U 27456,55700,27456,4872,512,36,45,729,8676,70434,265635,265635,70434,8676,729,45
%N Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.
%C Paths of length zero are not counted here.
%H Andrew Howroyd, <a href="/A360851/b360851.txt">Table of n, a(n) for n = 1..1275</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Induced_path">Induced path</a>.
%F T(m,n) = A360850(m,n) - A003991(m,n).
%F T(m,n) = -m*n + Sum_{j=1..min(m,n)} j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j).
%F T(m,n) = T(n,m).
%e Array begins:
%e ===================================================
%e m\n| 1 2 3 4 5 6 7 ...
%e ---+-----------------------------------------------
%e 1 | 0 1 3 6 10 15 21 ...
%e 2 | 1 8 27 64 125 216 343 ...
%e 3 | 3 27 126 426 1125 2493 4872 ...
%e 4 | 6 64 426 2208 8830 27456 70434 ...
%e 5 | 10 125 1125 8830 55700 265635 961975 ...
%e 6 | 15 216 2493 27456 265635 2006280 11158161 ...
%e 7 | 21 343 4872 70434 961975 11158161 98309778 ...
%e ...
%o (PARI) T(m,n) = sum(j=1, min(m,n), j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j)) - m*n
%Y Main diagonal is A360852.
%Y Rows 1..2 are A000217(n-1), A000578.
%Y Cf. A003991, A360199, A360850.
%K nonn,tabl
%O 1,4
%A _Andrew Howroyd_, Feb 24 2023
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