A187235 Number of ways to place n nonattacking semi-bishops on an n X n board.

1, 5, 51, 769, 15345, 381065, 11323991, 391861841, 15476988033, 687029386845, 33861652925595, 1834814222811361, 108411291759763681, 6936921762461326545, 477881176664541171375, 35264213540563039871265, 2775185864375851234241985, 232010235620834821000259765, 20534530616200868936398461635

Offset: 1

Author

Vaclav Kotesovec, Mar 08 2011

Keywords

• nonn
• nice

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..350
• V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 260-265.

Crossrefs

Cf. A238261, A238262, A002465, A099152, A000255, A129256.

Programs

Mathematica

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 *)

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

Formula

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).

For constants see A238261 and A238262. - Vaclav Kotesovec, Feb 21 2014

a(n) = (-1)^n * Sum_{i=0..n} Stirling1(n,i) * Stirling1(n+1,n-i+1). - Ryan Brooks, May 09 2020