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

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

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 and 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]
• Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
• Édouard 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)

Crossrefs

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

Programs

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

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 *)
With[{nn=30},CoefficientList[Series[(Exp[2x+x^2]+1)/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 27 2025 *)

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