

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.


6



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

0,2


COMMENTS

Series reversion of x(14x)/(12x).  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 nth term of this sequence is 2^(n1) times of the nth term of the big Schroeder number, Jan 28, 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 (unpublished)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, H. Pan, Identities involving weighted CatalanSchroder and Motzkin Paths, arXiv:1608.02448 (2016), eq. (1.13), a=2, b=4.
Ira M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28 (1980), 321337.
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*xsqrt(4*x^212*x+1))/(8*x).
From Paul Barry, May 19 2005: (Start)
a(n) = sum{k=0..n, C(n+1, k)C(2nk, n)(1)^k*4^(nk)*2^k}/(n+1);
a(n) = sum{k=0..n, (1/n)*C(n, k)*C(n, k+1)*4^k*2^(nk)};
a(n) = sum{k=1..n, N(n, k)*2^(n+k1)}, 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*n1)*a(n1)  4*(n2)*a(n2).  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*xsqrt(4*x^212*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*xSqrt[4*x^212*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*xsqrt(4*x^212*x+1))/(8*x)) \\ Joerg Arndt, May 06 2013


CROSSREFS

Cf. A006318, A001003, A054726.
Sequence in context: A193125 A305868 A319324 * A192621 A143923 A079858
Adjacent sequences: A059432 A059433 A059434 * A059436 A059437 A059438


KEYWORD

nonn,easy


AUTHOR

Wenjin Woan, Feb 01 2001


STATUS

approved



