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A191236
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Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.
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6
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1, 2, 14, 184, 3532, 89256, 2800016, 104967808, 4578528464, 227816059360, 12735645181536, 790296855912576, 53905019035510528, 4008716449677965312, 322807879692969879552, 27983800239966141382656
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 13 2012
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STATUS
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approved
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