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A187235
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Number of ways to place n nonattacking semi-bishops on an n X n board.
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20
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1, 5, 51, 769, 15345, 381065, 11323991, 391861841, 15476988033, 687029386845, 33861652925595, 1834814222811361, 108411291759763681, 6936921762461326545, 477881176664541171375, 35264213540563039871265, 2775185864375851234241985, 232010235620834821000259765, 20534530616200868936398461635
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OFFSET
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1,2
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COMMENTS
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Two semi-bishops do not attack each other if they are in the same NorthWest-SouthEast diagonal.
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
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LINKS
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FORMULA
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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).
a(n) = (-1)^n * Sum_{i=0..n} Stirling1(n,i) * Stirling1(n+1,n-i+1). - Ryan Brooks, May 09 2020
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MATHEMATICA
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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}]
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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