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

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.

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.

Sequence in context: A012882 A063017 A051643 this_sequence A129840 A085390 A065980

Adjacent sequences: A000888 A000889 A000890 this_sequence A000892 A000893 A000894

nonn

njas

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