A000899 - OEIS

%I M4645 N1987 #32 Jan 10 2018 16:04:50

%S 0,0,0,1,9,70,571,4820,44676,450824,4980274,59834748,778230060,

%T 10896609768,163456629604,2615335902176,44460874280032,

%U 800296440705472,15205636325496568,304112744618157872,6386367741011250672

%N Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

%D L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000899/b000899.txt">Table of n, a(n) for n = 1..200</a>

%H L. C. Larson, <a href="/A000900/a000900_1.pdf">The number of essentially different nonattacking rook arrangements</a>, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]

%H E. Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.

%H E. Lucas, <a href="/A000899/a000899.pdf">Théorie des nombres</a> (annotated scans of a few selected pages)

%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)

%F a(n)=(A000142(n)-2*A000085(n)-A037223(n)+2*A000898(floor(n/2)))/8 (all of which have explicit formulas).

%F For asymptotics see the Robinson paper.

%p For Maple program see A000903.

%t a[n_] := ((n+1)! - (2*Floor[(n+1)/2])!! - 2*Sum[Binomial[n+1, 2*k]*(2*k-1)!!, {k, 0, (n+1)/2}] + 2*Sum[2^k*BellB[k]*StirlingS1[Floor[(n+1)/2], k], {k, 0, Floor[(n+1)/2]}])/8; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 23 2013, from explicit formulas *)

%Y Cf. A000900.

%K nonn,nice,easy

%O 1,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, May 09 2000