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A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

%I A201862 #29 Feb 16 2025 08:33:16
%S A201862 1,2,9,70,729,9918,167281,3423362,82609921,2319730026,74500064809,
%T A201862 2711723081550,110568316431609,5016846683306758,251180326892449969,
%U A201862 13806795579059621930,827911558468860287041,53940895144894708523922,3799498445458163685753481,288400498147873552894868886
%N A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.
%C A201862 Also the number of vertex covers and independent vertex sets of the n X n bishop graph.
%H A201862 Vaclav Kotesovec, <a href="/A201862/b201862.txt">Table of n, a(n) for n = 0..320</a>
%H A201862 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>
%H A201862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H A201862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A201862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%F A201862 a(n) = A216078(n+1) * A216332(n+1). - _Andrew Howroyd_, May 08 2017
%t A201862 knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
%t A201862 *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 A201862 *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 A201862 Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]
%Y A201862 Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.
%Y A201862 Row sums of A378590.
%K A201862 nonn
%O A201862 0,2
%A A201862 _Vaclav Kotesovec_, Dec 06 2011
%E A201862 a(0)=1 prepended by _Alois P. Heinz_, Dec 01 2024