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A399048
Triangle read by rows: T(n,k) is the number of induced subgraphs of the n X n rook graph with k connected components.
3
1, 1, 1, 1, 13, 2, 1, 397, 108, 6, 1, 55933, 8690, 888, 24, 1, 31450861, 1950300, 145350, 7800, 120, 1, 67253507293, 1414836002, 48780600, 2277600, 74520, 720, 1, 559182556492477, 3722299038108, 43963422006, 1098031200, 35656320, 776160, 5040
OFFSET
0,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Eric Weisstein's World of Mathematics, Rook Graph.
FORMULA
A399044(n) = Sum_{k=0..n} k*T(n,k).
EXAMPLE
Triangle begins:
1;
1, 1;
1, 13, 2;
1, 397, 108, 6;
1, 55933, 8690, 888, 24;
1, 31450861, 1950300, 145350, 7800, 120;
...
PROG
(PARI) \\ In G: S is A183109, T is A262307 (compare A360873).
G(M, N=M)={ my(S=matrix(M, N), T=matrix(M, N), U=matrix(M, N), V=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]=y*(T[m, n] + sum(i=1, m-1, sum(j=1, n-1, T[i, j]*U[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j))));
V[m, n]=1+sum(i=1, m, sum(j=1, n, binomial(m, i)*binomial(n, j)*U[i, j])) ));
V
}
T(n)={my(A=G(n, n)); [Vecrev(p) | p<-concat([1], vector(n, i, A[i, i]))]}
{ my(A=T(7)); for(i=1, #A, print(A[i])) }
CROSSREFS
Main diagonal is A000142.
Row sums are A002416.
Columns k=0..1 are A000012, A286189.
Sequence in context: A010223 A010222 A086266 * A380392 A217436 A040166
KEYWORD
nonn,tabl,new
AUTHOR
Andrew Howroyd, Aug 18 2026
STATUS
approved