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A005632 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

%I A005632 M3923 #20 Jul 23 2022 19:26:25
%S A005632 0,0,5,22,258,1628,18052,145976,1837272,18407664,265312848,3184567136,
%T A005632 52020223648,728304073664,13317701313600,213083801827200,
%U A005632 4314950946864000,77669134543011584,1725980887361498368,34519618313219995136,835374767116711506432,18378244896208168541184
%N A005632 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).
%D A005632 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005632 R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence mu(2k+1).]
%H A005632 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 A005632 For Maple program see A005635.
%t A005632 c[n_] := Module[{k}, If[Mod[n, 2]==0, Return[0]]; k = (n-1)/2; If[Mod[k, 2] == 0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k*(((k+1)/2)!)^2]]];
%t A005632 d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
%t A005632 B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n-1], True, 2*B[n-2] + (n - 2)*B[n - 4]];
%t A005632 S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
%t A005632 Q[n_] := Module[{m}, If[Mod[n, 8] != 1, Return[0]]; m = (n-1)/8; ((2*m)!)^2 /(m!)^2];
%t A005632 a[n_] := (c[2n+1] - S[2n+1] - Q[2n+1])/4;
%t A005632 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Jul 23 2022, after Maple program in A005635 *)
%K A005632 nonn
%O A005632 1,3
%A A005632 _N. J. A. Sloane_
%E A005632 More terms from _N. J. A. Sloane_, Sep 28 2006