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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}. 5
 1, 3, 3, 6, 12, 6, 10, 33, 33, 10, 15, 72, 135, 72, 15, 21, 135, 438, 438, 135, 21, 28, 228, 1140, 2224, 1140, 228, 28, 36, 357, 2511, 8850, 8850, 2511, 357, 36, 45, 528, 4893, 27480, 55725, 27480, 4893, 528, 45, 55, 747, 8700, 70462, 265665, 265665, 70462, 8700, 747, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 Eric Weisstein's World of Mathematics, Complete Bipartite Graph. Eric Weisstein's World of Mathematics, Graph Path. Eric Weisstein's World of Mathematics, Rook Graph. FORMULA T(m,n) = Sum_{j=1..min(m,n)} j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j). T(m,n) = T(n,m). EXAMPLE Array begins: =================================================== m\n| 1 2 3 4 5 6 7 ... ---+----------------------------------------------- 1 | 1 3 6 10 15 21 28 ... 2 | 3 12 33 72 135 228 357 ... 3 | 6 33 135 438 1140 2511 4893 ... 4 | 10 72 438 2224 8850 27480 70462 ... 5 | 15 135 1140 8850 55725 265665 962010 ... 6 | 21 228 2511 27480 265665 2006316 11158203 ... 7 | 28 357 4893 70462 962010 11158203 98309827 ... ... PROG (PARI) T(m, n) = sum(j=1, min(m, n), j!^2*binomial(m, j)*binomial(n, j)*(1 + (m+n)/2 - j)) CROSSREFS Main diagonal is A288035. Rows 1..2 are A000217, A054602. Cf. A360849 (cycles), A360851. Sequence in context: A326498 A367644 A094305 * A360202 A057963 A250301 Adjacent sequences: A360847 A360848 A360849 * A360851 A360852 A360853 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Feb 23 2023 STATUS approved

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Last modified September 9 21:41 EDT 2024. Contains 375765 sequences. (Running on oeis4.)