A360873 Array read by antidiagonals: T(m,n) is the number of (non-null) connected induced subgraphs in the rook graph K_m X K_n.
1, 3, 3, 7, 13, 7, 15, 51, 51, 15, 31, 205, 397, 205, 31, 63, 843, 3303, 3303, 843, 63, 127, 3493, 27877, 55933, 27877, 3493, 127, 255, 14451, 233751, 943095, 943095, 233751, 14451, 255, 511, 59485, 1938517, 15678925, 31450861, 15678925, 1938517, 59485, 511
Offset: 1
Examples
Array begins: ======================================================= m\n| 1 2 3 4 5 6 ... ---+--------------------------------------------------- 1 | 1 3 7 15 31 63 ... 2 | 3 13 51 205 843 3493 ... 3 | 7 51 397 3303 27877 233751 ... 4 | 15 205 3303 55933 943095 15678925 ... 5 | 31 843 27877 943095 31450861 1033355223 ... 6 | 63 3493 233751 15678925 1033355223 67253507293 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals).
- Eric Weisstein's World of Mathematics, Rook Graph.
- Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph.
Crossrefs
Programs
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PARI
\\ S is A183109, T is A262307, U is this sequence. G(M,N=M)={ my(S=matrix(M, N), T=matrix(M, N), U=matrix(M, N)); for(m=1, M, for(n=1, N, S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n); 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))); U[m, n]=sum(i=1, m, sum(j=1, n, binomial(m, i)*binomial(n, j)*T[i, j])) )); U } { my(A=G(7)); for(n=1, #A~, print(A[n,])) }
Formula
T(m,n) = Sum_{i=1..m} Sum_{j=1..n} binomial(m, i) * binomial(n, j) * A262307(i, j).
T(m,n) = T(n,m).