|
|
A123072
|
|
Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).
|
|
5
|
|
|
1, 2, 72, 7200, 1411200, 457228800, 221298739200, 149597947699200, 134638152929280000, 155641704786247680000, 224746621711341649920000, 396453040698806670458880000, 838894634118674914690990080000, 2097236585296687286727475200000000, 6115541882725140128097317683200000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
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).]
|
|
FORMULA
|
From_Reinhard Zumkeller_, Feb 16 2010: (Start)
a(n) = ceiling((((2*n)! / n!)^2) / 2).
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) ~ 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
|
|
MAPLE
|
|
|
MATHEMATICA
|
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|