A151281 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (1, 0), (1, 1)}.

1, 2, 6, 16, 48, 136, 408, 1184, 3552, 10432, 31296, 92544, 277632, 824448, 2473344, 7365120, 22095360, 65920000, 197760000, 590790656, 1772371968, 5299916800, 15899750400, 47578857472, 142736572416, 427357700096, 1282073100288, 3840133464064, 11520400392192, 34517383151616, 103552149454848

Offset

0,2

Comments

From Paul Barry, Jan 26 2009: (Start)

Image of 2^n under A155761. Binomial transform is A129637. Hankel transform is 2^C(n+1,2).

In general, the g.f. of the reversion of x*(1+c*x)/(1+a*x+b*x^2) is given by the continued fraction x/(1 -(a-c)*x -(b-a*c+c^2)*x^2/(1 -(a-2*c)*x -(b-a*c+c^2)*x^2/(1 -(a-2*c)*x -(b-a*c+c^2)*x^2/(1 - .... (End)

a(n) is the number of nondeterministic Dyck meanders of length n. See A368164 or the de Panafieu-Wallner article for the definiton of nondeterministc walks. A nondeterministic meander contains at least one classical meander, i.e., a walk never crossing the x-axis. - Michael Wallner, Dec 18 2023

Links

Robert Israel, Table of n, a(n) for n = 0..2000

Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.

Elie De Panafieu, Mohamed Lamine Lamali, and Michael Wallner, Combinatorics of nondeterministic walks of the Dyck and Motzkin type, arXiv:1812.06650 [math.CO], 2018.

Élie de Panafieu and Michael Wallner, Combinatorics of nondeterministic walks, arXiv:2311.03234 [math.CO], 2023.

Formula

From Paul Barry, Jan 26 2009: (Start)

G.f.: 1/(1 -2*x -2*x^2/(1 -2*x^2/(1 -2*x^2/(1 -2*x^2/(1 -2*x^2/(1 - .... (continued fraction).

G.f.: c(2*x^2)/(1-2*x*c(2*x^2)) = (sqrt(1-8*x^2) + 4*x - 1)/(4*x*(1-3*x)).

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

Reversion of x*(1 + 2*x)/(1 + 4*x + 6*x^2). (End)

From Philippe Deléham, Feb 01 2009: (Start)

a(n) = Sum_{k=0..n} A120730(n,k)*2^k.

a(2*n+2) = 3*a(2*n+1), a(2*n+1) = 3*a(2*n) - 2^n*A000108(n).

a(2*n+1) = 3*a(2*n) - A151374(n). (End)

(n+1)*a(n) = 3*(n+1)*a(n-1) + 8*(n-2)*a(n-2) - 24*(n-2)*a(n-3). - R. J. Mathar, Nov 26 2012

a(n) ~ 3^n/2. - Vaclav Kotesovec, Feb 13 2014

Maple

N:= 1000: # to get terms up to a(N)
S:= series((sqrt(1-8*x^2)+4*x-1)/(4*x*(1-3*x)), x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, Feb 18 2013

Mathematica

aux[i_, j_, n_] := Which[Min[i, j, n]<0 || Max[i, j]>n, 0, n==0, KroneckerDelta[i, j, n], True, aux[i, j, n]= aux[-1+i, -1+j, -1+n] +aux[-1+i, j, -1+n] +aux[1+i, j, -1+n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
a[n_]:= a[n]= If[n<3, (n+1)!, (3*(n+1)*a[n-1] +8*(n-2)*a[n-2] -24*(n-2)*a[n-3])/(n+1)]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

Prog

(Magma) [n le 3 select Factorial(n) else (3*n*Self(n-1) + 8*(n-3)*Self(n-2) - 24*(n-3)*Self(n-3))/n: n in [1..41]]; // G. C. Greubel, Nov 09 2022
(SageMath)
def a(n): # a = A151281
    if (n==0): return 1
    elif (n%2==1): return 3*a(n-1) - 2^((n-1)/2)*catalan_number((n-1)/2)
    else: return 3*a(n-1)
[a(n) for n in (0..40)] # G. C. Greubel, Nov 09 2022

Crossrefs

Cf. A000108, A120730, A129637, A151374, A155761.

Cf. A368164 (nondeterministic Dyck bridges), A368234 (nondeterministic Dyck excursions).

Sequence in context: A148443 A148444 A064190 * A045694 A225178 A129772

Adjacent sequences: A151278 A151279 A151280 * A151282 A151283 A151284

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved