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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

References

• L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
• 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
• L. 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]
• E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
• E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Crossrefs

Cf. A000900.

Programs

Maple

For Maple program see A000903.

Mathematica

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 *)

Formula

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

For asymptotics see the Robinson paper.

Extensions

More terms from Vladeta Jovovic, May 09 2000