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A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.

%I A191236 #29 Mar 11 2022 11:42:25
%S A191236 1,2,14,184,3532,89256,2800016,104967808,4578528464,227816059360,
%T A191236 12735645181536,790296855912576,53905019035510528,4008716449677965312,
%U A191236 322807879692969879552,27983800239966141382656
%N A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.
%H A191236 G. C. Greubel, <a href="/A191236/b191236.txt">Table of n, a(n) for n = 0..340</a>
%H A191236 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 253.
%F A191236 a(n) = 1/n! * Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * (j*(j+1))^n.
%F A191236 Asymptotic: a(n) ~ 1/sqrt(Pi*(z-1)*(2-z)*n)*(2*n*exp(z-1)/z)^n or a(n) ~ exp(z/2)*Stirling2(2*n,n) where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2.
%F A191236 O.g.f.: Sum_{n>=0} n^n*(n+1)^n * exp(-n*(n+1)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n. - _Paul D. Hanna_, Oct 15 2012
%F A191236 a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(2*n-k,n), where Stirling2(n,k) = A008277(n,k). - _Paul D. Hanna_, Nov 13 2012
%t A191236 Join[{1}, Table[(1/n!)*Sum[(-1)^(n - k)*Binomial[n, k]*(k*(k + 1))^n, {k, 0, n}], {n,1,50}]] (* _G. C. Greubel_, Feb 03 2017 *)
%o A191236 (PARI) {a(n)=polcoeff(sum(m=0,n,m^m*(m+1)^m*x^m*exp(-m*(m+1)*x+x*O(x^n))/m!),n)} \\ _Paul D. Hanna_, Oct 15 2012
%o A191236 (PARI) {a(n)=sum(k=0,n, binomial(n,k) * stirling(2*n-k,n,2))} \\ _Paul D. Hanna_, Nov 13 2012
%Y A191236 Cf. A002465, A187235, A217905.
%K A191236 nonn
%O A191236 0,2
%A A191236 _Vaclav Kotesovec_, May 27 2011
%E A191236 Offset changed to 0 and a(0)=1 added by _Paul D. Hanna_, Nov 13 2012