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Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).
(Formerly M1964 N0777)
12

%I M1964 N0777 #32 Jan 10 2018 16:04:59

%S 0,0,0,1,2,10,28,106,344,1272,4592,17692,69384,283560,1191984,5171512,

%T 23087168,105883456,498572416,2404766224,11878871456,59975885856,

%U 309439708352,1628919330208,8746079933568,47840206525056

%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="/A000900/b000900.txt">Table of n, a(n) for n = 0..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)

%H R. G. Wilson, v, <a href="/A000900/a000900.pdf">Comments on the Larsen paper (no date)</a>

%F a(n)=(A000085(n)-A000898(int(n/2)))/2

%F For asymptotics see the Robinson paper.

%p For Maple program see A000903.

%t a85[n_] := Sum[ (2k)!/k!/2^k Binomial[n, 2k], {k, 0, n/2}]; a898[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; a[n_] := (a85[n] - a898[Floor[n/2]])/2; a[1] = 0; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 13 2011, after formula *)

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, May 09 2000