A368234 -id:A368234 - cp's OEIS frontend

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

Manuel Kauers, Nov 18 2008

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

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.

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

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

[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

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

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 *)

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

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

A368164 Number of nondeterministic Dyck bridges of length 2*n.

Original entry on oeis.org

1, 7, 63, 583, 5407, 50007, 460815, 4231815, 38745279, 353832631, 3224323183, 29328492519, 266364307231, 2416023142423, 21890268365007, 198151683934023, 1792260214473087, 16199857938091383, 146342491104098607, 1321339563995562663, 11925412051760977887, 107590261672922633943

Offset: 0

Michael Wallner, Dec 14 2023

nonn

In nondeterministic walks (N-walks) the steps are sets and called N-steps. N-walks start at 0 and are concatenations of such N-steps such that all possible extensions are explored in parallel. The nondeterministic Dyck step set is { {-1}, {1}, {-1,1} }. Such an N-walk is called an N-bridge if it contains at least one trajectory that is a classical bridge, i.e., starts and ends at 0 (for more details see the de Panafieu-Wallner article).

			The a(1)=7 N-bridges of length 2 are
           /              /    /
/\,    ,  /\,    ,  /\,  / ,  /\
     \/        \/   \    \/   \/
                \    \         \

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

Cf. A151281 (nondeterministic Dyck meanders), A368234 (nondeterministic Dyck excursions), A000244 (nondeterministic Dyck walks).

G.f.: (1-6*t)/(sqrt(1-8*t)*(1-9*t)).
From Joseph M. Shunia, May 09 2024: (Start)
a(n) = A089022(n) + Sum_{k=0..n-1} binomial(2*n, k)*2^(2*n-k).
a(n) = A000244(2*n) - Sum_{k=n+1..2*n} binomial(2*n, k)*2^(2*n-k+1). (End)

cp's OEIS Frontend

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)}.

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