|
| |
|
|
A000891
|
|
(2*n)!*(2*n+1)! / (n! * (n+1)!)^2.
|
|
18
| |
|
|
1, 3, 20, 175, 1764, 19404, 226512, 2760615, 34763300, 449141836, 5924217936, 79483257308, 1081724803600, 14901311070000, 207426250094400, 2913690606794775, 41255439318353700, 588272005095043500
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Number of parallelogram polyominoes having n+1 columns and n+1 rows. - Emeric Deutsch, 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, 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, Oct 14 2008]
For n>0, A000891(n)/(n+2) = A000356 starting (1, 5, 35, 294,...). [Gary W. Adamson, Apr 8 2011]
|
|
|
REFERENCES
| 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.
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..100
|
|
|
FORMULA
| G.f.: (1 - E(16*x)/(pi/2))/(4*x) where E is the elliptic integral of the second kind.
G.f.: 3F2(1, 1/2, 3/2; 2,2; 16*x)= (1 - 2F1(-1/2, 1/2; 1; 16*x))/(4*x).
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)*BesselI(1, 2*x)/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).
G.f.: If G_N(x)=1+sum('((2*k)!*(2*k+1)!*x^k)/(((k!)*((k+1)!))^2)', 'k'=1..N), G_N(x)=1+12*x/(G(0)-12*x); G(k)=16*x*(k^2)+32*x*k+(k^2)+4*k+12*x+4-4*x*(2*k+3)*(2*k+5)*((k+2)^2)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
|
|
|
MAPLE
| 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
|
|
|
PROG
| (PARI) a(n)=binomial(2*n+1, n)^2/(2*n+1) /* Michael Somos Jun 22 2005 */
(MAGMA) [Factorial(2*n)*Factorial(2*n+1) / (Factorial(n) * Factorial(n+1))^2: n in [0..20]]; // Vincenzo Librandi, Aug 15 2011
|
|
|
CROSSREFS
| A010370(n+1)=-4a(n).
Cf. A038535.
A145600, A145601, A145602, A145603. [From Peter Bala, Oct 14 2008]
Cf. A000356
Sequence in context: A145329 A051643 A154644 * A129840 A085390 A065980
Adjacent sequences: A000888 A000889 A000890 * A000892 A000893 A000894
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 24 2008
Legend for G.f. formula plus another hypergeometric variation from Olivier Gérard (olivier.gerard(AT)gmail.com), Feb 16 2011
|
| |
|
|
