OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
FORMULA
a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i)*binomial(n,j)*A262307(i,j). - Andrew Howroyd, May 22 2017
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 12 2017
MATHEMATICA
{1} ~ Join ~ Table[g = GraphData[{"Rook", {n, n}}]; -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[n^2]}], {n, 2, 4}]
(* Second program: *)
b[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}];
T[m_, n_] := T[m, n] = b[m, n] - Sum[T[i, j]*b[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
a[n_] := Sum[Binomial[n, i]*Binomial[n, j]*T[i, j], {i, 1, n}, {j, 1, n}];
Array[a, 12] (* Jean-François Alcover, Oct 11 2017, after Andrew Howroyd *)
PROG
(PARI)
G(N)={my(S=matrix(N, N), T=matrix(N, N), U=matrix(N, N));
for(m=1, N, 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}
a(n)=G(n)[n, n]; \\ Andrew Howroyd, May 22 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, May 04 2017
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, May 22 2017
STATUS
approved