A238262 -id:A238262 - cp's OEIS frontend

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

Vaclav Kotesovec, Mar 08 2011

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

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

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

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 *)

a(n) = {(-1)^n*sum(i=0, n, stirling(n,i,1) * stirling(n+1,n-i+1,1))} \\ Andrew Howroyd, May 09 2020

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

A238261 Decimal expansion of a constant related to A187235.

Original entry on oeis.org

4, 9, 1, 0, 8, 1, 4, 9, 6, 4, 5, 6, 8, 2, 5, 5, 8, 9, 8, 7, 5, 1, 5, 3, 4, 8, 0, 5, 2, 4, 0, 3, 5, 2, 1, 9, 7, 8, 9, 8, 7, 0, 5, 2, 8, 1, 7, 6, 7, 8, 4, 7, 1, 7, 6, 1, 3, 9, 4, 1, 1, 2, 0, 2, 2, 5, 6, 4, 1, 7, 8, 7, 7, 8, 7, 9, 9, 4, 7, 9, 7, 2, 9, 5, 1, 8, 1, 9, 7, 4, 1, 5, 3, 5, 5, 4, 4, 6, 1, 4, 2, 5, 0, 5, 3

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			4.9108149645682558987515348...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238262, A002465, A129256, A342111, A330287.

RealDigits[N[-(2*LambertW[-1,-1/2/Sqrt[E]])^2/(1+2*LambertW[-1,-1/2/Sqrt[E]]), 105]][[1]]

Equals lim n->infinity (A187235(n)/(n-1)!)^(1/n).

Equals -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)).

cp's OEIS Frontend

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

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