1,3

Also, number of even minus number of odd extensions of truncated n-1 by n grid diagram.

Also, a(n) is the degree of the spinor variety, the complex projective variety SO(2n+1)/U(n). See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002

Also, number of ways of placing 1,...,n in a triangular array such that both rows and columns are increasing. - Jon Perry (perry(AT)globalnet.co.uk), Jun 13 2003

E.g. a(3)=2 as we can write:

1

23

456

or

1

24

356

Also, the number of symbolic sequences on the n symbols 0,1, ..., n-1. A symbolic sequence is a sequence that has n occurrences of 0, n-1 occurrences of 1, ..., one occurrence of n-1 and satisfies the condition that between any two consecutive occurrences of the symbol i it has exactly one occurrence of the symbol i+1. For example, the two symbolic sequences on 3 symbols are 012010 and 010210. The Shapiro-Shapiro paper shows how such sequences arise in the study of the arrangement of the real roots of a polynomial and its derivatives. There is a natural bijection between symbolic sequences and the triangular arrays described above. - Peter Bala (pbala(AT)toucansurf.com), Jul 18 2007

a(n) also appears to be the number of chains from w_0 down to 1 in a certain suborder of the strong Bruhat order on S_n: we let v cover (ij)*v only if i,j straddle the leftmost descent in v. For n=3 and drawing that descent with a |, this order is 3|21 > 23|1 > 13|2 & 2|13 > 123, with two maximal chains. [From Allen Knutson (allenk(AT)math.cornell.edu), Oct 13 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.

H. Hiller. Combinatorics and intersection of Schubert varieties. Comment. Math. Helv. 57 (1982), 41-59.

G. Kreweras, Sur un probleme de scrutin a plus de deux candidats, Publications de l'Institut de Statistique de l'Universit\'{e} de Paris, 26 (1981), 69-87.

R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.

F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.

B. Shapiro and M. Shapiro, A few riddles behind Rolle's theorem

R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

Dennis White, Sign-balanced posets

C(n, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!)

(PARI) a(n)=((n*n+n)/2)!*prod(i=1, n, (i-1)!/(2*i-1)!)

Cf. A005118, A018241, A007724, A004065.

Cf. A131811.

Adjacent sequences: A003118 A003119 A003120 this_sequence A003122 A003123 A003124

Sequence in context: A012444 A012754 A083568 this_sequence A057170 A008338 A000178

nonn,nice,easy

C. L. Mallows (colinm(AT)research.avayalabs.com)

More terms from Michael Somos . Additional references from Frank Ruskey (fruskey(AT)cs.uvic.ca)