0,2

Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 21 2003

Number of tilings of a <n,2,n> hexagon.

a(n) = number of non-crossing partitions of [2n+1] into n+1 blocks. For example, a[1] counts 13-2, 1-23, 12-3. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005

a(n)=A001700(n)*A000108(n) =(1/2)*A000984(n+1)*A000108(n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2007

The number of returning walks of length 2n on the upper half of a square lattice, since a(n)=Sum_{k=0..2n}Binomial(2n,k)A126120(k)A126869(n-k). - Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008

For sequences counting walks in the upper half-plane starting from the origin and finishing at the lattice points (0,m) see A145600 (m = 1), A145601 (m = 2), A145602 (m = 3) and A145603 (m = 4). [From Peter Bala (pbala(AT)toucansurf.com), Oct 14 2008]

E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.

W. Y. C. Chen, S. X. M. Pang, E. X. Y. Qu and R. P Stanley, Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions, http://arxiv.org/abs/0804.2930; Discrete Math., 309 (2009), 2834-2838.

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 94.

G.f.: (1 - E(16x)/(pi/2))/(4x) = (1 - F(-1/2, 1/2; 1; 16x))/(4x).

E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(0, 2x)*BesselI(1, 2x)/x . - Michael Somos Jun 22 2005

a(n) = A001263(2*n+1,n+1) = binomial(2*n+1,n+1)*binomial(2*n+1,n)/(2*n+1) (central members of odd numbered rows of Narayana triangle).

with(combstruct): bin := {B=Union(Z, Prod(B, B))} :seq(1/2*binomial(2*i, i)*(count([B, bin, unlabeled], size=i)), i=1..18) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2007

(PARI) a(n)=binomial(2*n+1, n)^2/(2*n+1) /* Michael Somos Jun 22 2005 */

A010370(n+1)=-4a(n).

Cf. A038535.

A145600, A145601, A145602, A145603. [From Peter Bala (pbala(AT)toucansurf.com), Oct 14 2008]

Sequence in context: A145329 A051643 A154644 this_sequence A129840 A085390 A065980

Adjacent sequences: A000888 A000889 A000890 this_sequence A000892 A000893 A000894

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008