cp's OEIS Frontend

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

%I A123072 #29 Dec 04 2022 08:32:25
%S A123072 1,2,72,7200,1411200,457228800,221298739200,149597947699200,
%T A123072 134638152929280000,155641704786247680000,224746621711341649920000,
%U A123072 396453040698806670458880000,838894634118674914690990080000,2097236585296687286727475200000000,6115541882725140128097317683200000000
%N A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).
%H A123072 R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence zeta(2k+1).]
%F A123072 From_Reinhard Zumkeller_, Feb 16 2010: (Start)
%F A123072 a(n) = ceiling((((2*n)! / n!)^2) / 2).
%F A123072 a(n) = A001700(n-1) * A010050(n). (End)
%F A123072 From _Benedict W. J. Irwin_, Jun 05 2016: (Start)
%F A123072 G.f. for a(n)/(n!)^2 : 1/2 + EllipticK(16*x)/Pi, which is the E.g.f for A187535.
%F A123072 G.f. for a(n)/(n!)^3 : 2F2(1/2, 1/2; 1, 1; 16z)/2.
%F A123072 a(n) = n!*A187535(n) = binomial(2*n-1, n-1)*(2*n)!.
%F A123072 (End)
%F A123072 a(n) = A156992(2n,n). - _Alois P. Heinz_, Apr 30 2017
%F A123072 a(n) ~ asy(2*n-1) where asy(n) = (2*n/e)^n*(18*n + 6 + 1/n)/9. - _Peter Luschny_, Dec 05 2019
%F A123072 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
%p A123072 For Maple program see A005635.
%t A123072 Table[(((2 n)!/n!)^2)/2, {n, 1, 20}] (* _Benedict W. J. Irwin_, Jun 05 2016 *)
%t A123072 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 *)
%Y A123072 Cf. A173331. [_Reinhard Zumkeller_, Feb 16 2010]
%Y A123072 Cf. A001700, A010050, A156992, A187535.
%K A123072 nonn
%O A123072 0,2
%A A123072 _N. J. A. Sloane_, Sep 28 2006
%E A123072 a(0)=1 prepended by _Alois P. Heinz_, Apr 30 2017