A000902 Expansion of e.g.f. (1/2)*(exp(2*x + x^2) + 1). (Formerly M2853 N1147)

1, 1, 3, 10, 38, 156, 692, 3256, 16200, 84496, 460592, 2611104, 15355232, 93376960, 585989952, 3786534784, 25152768128, 171474649344, 1198143415040, 8569374206464, 62668198184448, 468111364627456, 3568287053001728

Offset

0,3

Comments

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

References

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).

Links

T. D. Noe, Table of n, a(n) for n=0..200

Aram Bingham, Özlem Uğurlu, Sects, rooks, pyramids, partitions and paths for type DIII clans, arXiv:1907.08875 [math.CO], 2019.

Aram Bingham and Özlem Uğurlu, DIII clan combinatorics for the orthogonal Grassmannian, Australasian J. of Combinatorics (2021) Vol. 79, No. 1, 55-86.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]

E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.

E. Lucas, Théorie des nombres (annotated scans of a few selected pages)

B. Pittel, Where the typical set partitions meet and join, Electron. J. of Combin. 7, R5.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Formula

a(n) = 2*a(n-1) + (2n-2)*a(n-2) for n >= 3. - N. J. A. Sloane, Sep 23 2006

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

Maple

# Comment from the authors: For Maple program see A000903.
A000902 := n -> `if`(n=0, 1, I^(-n)*orthopoly[H](n, I)/2):
seq(A000902(n), n=0..22); # Peter Luschny, Nov 29 2017

Mathematica

n = 22; CoefficientList[ Series[(1/2)*(Exp[2*x+x^2] + 1), {x, 0, n}], x] * Table[k!, {k, 0, n}]
(* Jean-François Alcover, May 18 2011 *)

Prog

(Haskell)
a000902 n = a000902_list !! n
a000902_list = 1 : 1 : 3 : map (* 2) (zipWith (+)
   (drop 2 a000902_list) (zipWith (*) [2..] $ tail a000902_list))
-- Reinhard Zumkeller, Sep 10 2013
(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

Crossrefs

Equals 1/2 * A000898(n) for n>0.

Sequence in context: A259690 A001002 A151062 * A151063 A103138 A074527

Adjacent sequences: A000899 A000900 A000901 * A000903 A000904 A000905

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Simon Plouffe

Status

approved