A059435 Number of lattice paths in plane starting at (0,0) and ending at (n,n) with steps from {(i,j): i+j > 0, i, j >= 0} that never go below the line y = x.

1, 2, 12, 88, 720, 6304, 57792, 547712, 5323008, 52761088, 531311616, 5420488704, 55905767424, 581954543616, 6106210615296, 64513688174592, 685741070942208, 7328106153115648, 78684992821788672, 848487859401261056

Offset

0,2

Comments

Series reversion of x(1-4x)/(1-2x). - Paul Barry, May 19 2005

The Hankel transform of this sequence is 8^C(n+1,2) = [1, 8, 512, 262144, ...]. - Philippe Deléham, Nov 08 2007

References

W.-J. Woan, A bijective proof by induction that the n-th term of this sequence is 2^(n-1) times of the n-th term of the big Schroeder number, 2001 (unpublished).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

David Callan, A uniformly distributed statistic on a class of lattice paths, Electronic J. Combinatorics, Vol. 11(1), R82, 2004.

Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 (2016); see Eq. (1.13) with a=2 and b=4.

Ira M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28 (1980), 321-337.

Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020. See B(2,n) column in Table 3 (p. 10).

Robert A. Sulanke, Counting lattice paths by Narayana polynomials, Electronic J. Combinatorics 7 (2000), R40.

Formula

a(n) = 2^n*A001003(n).

G.f.: (1 + 2*x - sqrt(4*x^2 - 12*x + 1))/(8*x).

From Paul Barry, May 19 2005: (Start)

a(n) = (1/(n + 1)) * Sum_{k=0..n} C(n+1, k) * C(2*n-k, n)(-1)^k * 4^(n-k) * 2^k;

a(n) = Sum_{k=0..n} (1/n) * C(n, k) * C(n, k+1) * 4^k * 2^(n-k);

a(n) = Sum_{k=1..n} N(n, k)*2^(n+k-1), for n >= 1, where N(n, k) are the Narayana numbers (A001263). [Corrected by Alejandro H. Morales, May 14 2015]

(End)

Recurrence: (n+1)*a(n) = 6*(2*n-1)*a(n-1) - 4*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 11 2012

a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012

Maple

gf := (1+2*x-sqrt(4*x^2-12*x+1))/(8*x): s := series(gf, x, 100): for i from 0 to 50 do printf(`%d, `, coeff(s, x, i)) od:

Mathematica

Table[SeriesCoefficient[(1+2*x-Sqrt[4*x^2-12*x+1])/(8*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 11 2012 *)

Prog

(PARI) x='x+O('x^66); Vec((1+2*x-sqrt(4*x^2-12*x+1))/(8*x)) \\ Joerg Arndt, May 06 2013

Crossrefs

Cf. A001003, A006318, A054726.

Sequence in context: A305868 A356830 A319324 * A192621 A372090 A143923

Adjacent sequences: A059432 A059433 A059434 * A059436 A059437 A059438

Keyword

nonn,easy

Author

Wenjin Woan, Feb 01 2001

Status

approved