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A123072
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Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).
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5
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1, 2, 72, 7200, 1411200, 457228800, 221298739200, 149597947699200, 134638152929280000, 155641704786247680000, 224746621711341649920000, 396453040698806670458880000, 838894634118674914690990080000, 2097236585296687286727475200000000, 6115541882725140128097317683200000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..14.
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).]
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FORMULA
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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
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MAPLE
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For Maple program see A005635.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A173331. [Reinhard Zumkeller, Feb 16 2010]
Cf. A001700, A010050, A156992, A187535.
Sequence in context: A253316 A051443 A246486 * A351764 A317346 A099681
Adjacent sequences: A123069 A123070 A123071 * A123073 A123074 A123075
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Sep 28 2006
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EXTENSIONS
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a(0)=1 prepended by Alois P. Heinz, Apr 30 2017
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STATUS
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approved
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