Search: id:A100070
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%I A100070
%S A100070 6,117,5632,515625,77262336,17230990189,5360119185408,2219048868131217,
%T A100070 1180000000000000000,783948341202404638821,636404158746280870281216,
%U A100070 619884903445287035295372217,713552333492738487958741450752
%N A100070 Number a(n) of forests with two components in the complete bipartite 
               graph K_{n,n}.
%C A100070 This sequence (a(n)) appears to dominate the sequence (n^{2n-2}) of the 
               number of spanning trees in K_{n,n} for n>1. This shows that the 
               sequence of independent set numbers for the cycle matroid of K_{n,
               n} is not monotone increasing unlike the complete graph K_{n}.
%D A100070 N. Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs, preprint, 
               2004.
%F A100070 a(n)=2(n^{2}-n))^{n-1}+(1/2!) sum_{x, y\in [n-1]}b(n, x, y), where b(n, 
               x, y)=binom{n}{x} binom{n}{y}x^{y-1}y^{x-1}(n-x)^{n-y-1}(n-y)^{n-x-1}
%e A100070 a(2)=6 because K_{2,2} is C_{4} the cycle of length 4 and there are 6 
               forests with two components in C_{4}.
%t A100070 a[n_]:=Sum[Binomial[n, x]*Binomial[n, y]*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1), 
               {x, 1, n-1}, {y, 1, n-1}]/2 + (2*(n^2-n)^(n-1)); Table[a[n], {n, 
               2, 10}] (* This will generate a(n) from n=2 to 10. *)
%Y A100070 Cf. A069087, A083483, A000272.
%Y A100070 Sequence in context: A052465 A113015 A024275 this_sequence A135869 A054957 
               A081537
%Y A100070 Adjacent sequences: A100067 A100068 A100069 this_sequence A100071 A100072 
               A100073
%K A100070 nonn
%O A100070 2,1
%A A100070 Woong Kook (andrewk(AT)math.uri.edu), Nov 02 2004

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