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 A290586 Number of irredundant sets in the n X n rook graph. 5
 2, 11, 94, 1185, 20106, 453271, 13169346, 476777153, 20869990066, 1076251513071, 64077661097418, 4337014196039377, 329768528011095642, 27905789218764082151, 2608140451597365915346, 267506385903592339178241, 29943760423790270319833826 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..100 Andrew Howroyd, Irredundant Sets in Rook Graphs Eric Weisstein's World of Mathematics, Irredundant Set Eric Weisstein's World of Mathematics, Rook Graph FORMULA a(n) = 2*n^n - n! + Sum_{k=0..n-1} Sum_{r=2*k..n-1} binomial(n,k) * binomial(n,r) * k! * A008299(r,k) * c(n-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)). - Andrew Howroyd, Aug 11 2017 MATHEMATICA s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}]; c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}]; p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k, 0, m - 1}]; Table[2*n^n - n! + p[n, n, 1], {n, 30}] (* Indranil Ghosh, Aug 12 2017, after PARI code *) PROG (PARI) \\ here s(n, k) is A008299, 2*n^n - n! is A248744. s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) ); c(m, n, x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i); p(m, n, x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m, k) * binomial(n, r) * k! * s(r, k) * x^r * c(m-k, n-r, x) ))} a(n) = 2*n^n - n! + p(n, n, 1); \\ Andrew Howroyd, Aug 11 2017 CROSSREFS Main diagonal of A290818. Row sums of A290823. Cf. A008299, A248744. Sequence in context: A158837 A236962 A292916 * A098621 A266834 A131407 Adjacent sequences: A290583 A290584 A290585 * A290587 A290588 A290589 KEYWORD nonn AUTHOR Eric W. Weisstein, Aug 07 2017 EXTENSIONS a(4) corrected and a(5) from Andrew Howroyd, Aug 07 2017 Terms a(6) and beyond from Andrew Howroyd, Aug 11 2017 STATUS approved

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Last modified September 8 22:43 EDT 2024. Contains 375759 sequences. (Running on oeis4.)