A000899 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details). (Formerly M4645 N1987)

0, 0, 0, 1, 9, 70, 571, 4820, 44676, 450824, 4980274, 59834748, 778230060, 10896609768, 163456629604, 2615335902176, 44460874280032, 800296440705472, 15205636325496568, 304112744618157872, 6386367741011250672, 140500090807262405024, 3231502091535943714096, 77556050212036641134016

Offset

1,5

References

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..200

Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]

Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.

Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages)

Robert W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976); Annotated scanned copy.

Formula

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

a(n) ~ (1/4) * sqrt(Pi*n/2) * (n/e)^n (Robinson, 1976, p. 202). - Amiram Eldar, Jul 18 2026

Maple

# For Maple program see A000903.

Mathematica

a[n_] := (n! - (2*Floor[n/2])!! - 2*Sum[Binomial[n, 2*k]*(2*k - 1)!!, {k, 0, n/2}] + 2*Sum[2^k*BellB[k]*StirlingS1[Floor[n/2], k], {k, 0, Floor[n/2]}])/8; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Dec 23 2013, from explicit formulas *)

Crossrefs

Cf. A000085, A000142, A000898, A000900, A000903, A037223.

Sequence in context: A045739 A098205 A383204 * A226013 A382531 A156705

Adjacent sequences: A000896 A000897 A000898 * A000900 A000901 A000902

Keyword

nonn,nice,easy,changed

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, May 09 2000

Status

approved