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%I #54 May 09 2020 19:11:53
%S 1,5,51,769,15345,381065,11323991,391861841,15476988033,687029386845,
%T 33861652925595,1834814222811361,108411291759763681,
%U 6936921762461326545,477881176664541171375,35264213540563039871265,2775185864375851234241985,232010235620834821000259765,20534530616200868936398461635
%N Number of ways to place n nonattacking semi-bishops on an n X n board.
%C Two semi-bishops do not attack each other if they are in the same NorthWest-SouthEast diagonal.
%C Conjecture: Number of parity preserving permutations of the set {1, 2, ..., 2n+1} with exactly n+1 cycles (see A246117). - _Peter Luschny_, Feb 09 2015
%H Vaclav Kotesovec, <a href="/A187235/b187235.txt">Table of n, a(n) for n = 1..350</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 260-265.
%F a(n)/(n-1)! ~ 0.24252191 * 4.9108149^n where the second constant is 1/(z*(1-z)) = 4.910814964..., where z=0.715331862959... is a root of the equation z=2*(z-1)*log(1-z).
%F For constants see A238261 and A238262. - _Vaclav Kotesovec_, Feb 21 2014
%F a(n) = (-1)^n * Sum_{i=0..n} Stirling1(n,i) * Stirling1(n+1,n-i+1). - _Ryan Brooks_, May 09 2020
%t Table[If[n==1,1,Coefficient[Expand[Product[x+i,{i,1,n}]*Product[x+i,{i,1,n-1}],x],x,n-1]],{n,1,50}]
%t Table[(-1)^n*Sum[StirlingS1[n+1,j]*StirlingS1[n,n-j+1],{j,1,n}],{n,1,50}] (* Explicit formula, _Vaclav Kotesovec_, Mar 24 2011 *)
%o (PARI) a(n) = {(-1)^n*sum(i=0, n, stirling(n,i,1) * stirling(n+1,n-i+1,1))} \\ _Andrew Howroyd_, May 09 2020
%Y Cf. A238261, A238262, A002465, A099152, A000255, A129256.
%K nonn,nice
%O 1,2
%A _Vaclav Kotesovec_, Mar 08 2011