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A000902
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Expansion of e.g.f. (1/2)*(exp(2*x + x^2) + 1).
(Formerly M2853 N1147)
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6
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1, 1, 3, 10, 38, 156, 692, 3256, 16200, 84496, 460592, 2611104, 15355232, 93376960, 585989952, 3786534784, 25152768128, 171474649344, 1198143415040, 8569374206464, 62668198184448, 468111364627456, 3568287053001728
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OFFSET
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0,3
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COMMENTS
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Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
One more than the number of ordered pairs of minimally intersecting partitions such that p consists of exactly two blocks.
The number of B-orbits in the symmetric space of type DIII, SO_{2n}(C)/GL_n(C) where B is a Borel subgroup of SO_{2n}(C). These are parameterized by "type DIII (n,n)-clans". E.g., for n=2, the a(2)=3 type DIII (2,2)-clans are ++--, --++, and 1212. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020
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REFERENCES
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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).
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LINKS
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R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
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FORMULA
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a(n) = 1 + n!/(2e) * [x^n] Sum[l>=0, 1/l! * {(1+x)^l-1}^2].
For asymptotics see the Robinson paper.
But the asymptotic formula in the Robinson paper is wrong (see A000898, discussion from Oct 01 2013). - Vaclav Kotesovec, Aug 04 2014
a(n) ~ 2^(n/2-3/2) * n^(n/2) * exp(sqrt(2*n)-n/2-1/2). - Vaclav Kotesovec, Aug 04 2014
a(n) = (i/2)^(1 - n)*KummerU((1 - n)/2, 3/2, -1)) for n>=1. - Peter Luschny, Nov 29 2017
a(n) = Sum_{r=0..floor(n/2)} 2^(n-2r-1) * {(n!)/(r!(n-2r)!)}. - Aram Bingham, Feb 08 2020
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MAPLE
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# Comment from the authors: For Maple program see A000903.
A000902 := n -> `if`(n=0, 1, I^(-n)*orthopoly[H](n, I)/2):
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MATHEMATICA
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n = 22; CoefficientList[ Series[(1/2)*(Exp[2*x+x^2] + 1), {x, 0, n}], x] * Table[k!, {k, 0, n}]
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PROG
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(Haskell)
a000902 n = a000902_list !! n
a000902_list = 1 : 1 : 3 : map (* 2) (zipWith (+)
(drop 2 a000902_list) (zipWith (*) [2..] $ tail a000902_list))
(Magma) a:=[1, 3]; [1] cat [n le 2 select a[n] else 2*Self(n-1) + (2*n-2)*Self(n-2):n in [1..22]]; // Marius A. Burtea, Feb 12 2020
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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