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Bishops on an n X n board (see Robinson paper for details).
1

%I #8 Jul 23 2022 19:25:45

%S 1,1,2,2,4,8,16,40,72,260,432,1976,2880,17632,23040,177248,201600,

%T 2001680,2016000,24879520,21772800,338969216,261273600,5002865792,

%U 3353011200,79676972608,46942156800,1358997441920,697426329600,24740358817280,11158821273600,478218277674496

%N Bishops on an n X n board (see Robinson paper for details).

%D R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (M_n, p. 208)

%H R. W. Robinson, <a href="/A000899/a000899_1.pdf">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

%p unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085

%p M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end;

%t d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];

%t a[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[k!*(k + 2)/2], Return[(k - 1)!*(k + 1)^2/2]], k = (n - 1)/2; Return[d[k]*d[k + 1]]]];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jul 23 2022, after Maple code *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 25 2006