A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.
1, 2, 14, 184, 3532, 89256, 2800016, 104967808, 4578528464, 227816059360, 12735645181536, 790296855912576, 53905019035510528, 4008716449677965312, 322807879692969879552, 27983800239966141382656
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..340
- Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 253.
Programs
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Mathematica
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 *) -
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 -
PARI
{a(n)=sum(k=0,n, binomial(n,k) * stirling(2*n-k,n,2))} \\ Paul D. Hanna, Nov 13 2012
Formula
a(n) = 1/n! * Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * (j*(j+1))^n.
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.
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
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
Extensions
Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 13 2012