|
%I #23 Aug 24 2020 11:56:27
%S 2,9,70,729,9918,167281,3423362,82609921,2319730026,74500064809,
%T 2711723081550,110568316431609,5016846683306758,251180326892449969,
%U 13806795579059621930,827911558468860287041,53940895144894708523922,3799498445458163685753481,288400498147873552894868886
%N Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.
%C Also the number of vertex covers and independent vertex sets of the n X n bishop graph.
%H Vaclav Kotesovec, <a href="/A201862/b201862.txt">Table of n, a(n) for n = 1..320</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%F a(n) = A216078(n+1) * A216332(n+1). - _Andrew Howroyd_, May 08 2017
%t knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
%t *Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,1,n-i}]
%t *Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,1,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);
%t Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]
%Y Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.
%K nonn
%O 1,1
%A _Vaclav Kotesovec_, Dec 06 2011
|
