A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).

1, 2, 72, 7200, 1411200, 457228800, 221298739200, 149597947699200, 134638152929280000, 155641704786247680000, 224746621711341649920000, 396453040698806670458880000, 838894634118674914690990080000, 2097236585296687286727475200000000, 6115541882725140128097317683200000000

Offset: 0

Author

N. J. A. Sloane, Sep 28 2006

Keywords

• nonn

Links

• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence zeta(2k+1).]

Crossrefs

Cf. A173331. [Reinhard Zumkeller, Feb 16 2010]

Cf. A001700, A010050, A156992, A187535.

Programs

Maple

For Maple program see A005635.

Mathematica

Table[(((2 n)!/n!)^2)/2, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)
Table[SeriesCoefficient[Series[1/2 + EllipticK[16 x]/Pi, {x, 0, 20}],n] n! n!, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)

Formula

From_Reinhard Zumkeller_, Feb 16 2010: (Start)

a(n) = ceiling((((2*n)! / n!)^2) / 2).

a(n) = A001700(n-1) * A010050(n). (End)

From Benedict W. J. Irwin, Jun 05 2016: (Start)

G.f. for a(n)/(n!)^2 : 1/2 + EllipticK(16*x)/Pi, which is the E.g.f for A187535.

G.f. for a(n)/(n!)^3 : 2F2(1/2, 1/2; 1, 1; 16z)/2.

a(n) = n!*A187535(n) = binomial(2*n-1, n-1)*(2*n)!.

(End)

a(n) = A156992(2n,n). - Alois P. Heinz, Apr 30 2017

a(n) ~ asy(2*n-1) where asy(n) = (2*n/e)^n*(18*n + 6 + 1/n)/9. - Peter Luschny, Dec 05 2019

Sum_{n>=0} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 30 2017