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A151374 Number of walks within N^2 (the first quadrant of Z^2) starting at (0, 0), ending on the vertical axis and consisting of 2n steps taken from {(-1, -1), (-1, 0), (1, 1)}. 23
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920, 51309480813527040, 383705682605506560, 2877792619541299200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A052701 shifted one place left. - R. J. Mathar, Dec 13 2008

Expansion of c(2*x), where c(x) is the g.f. of A000108. - Philippe Deléham, Feb 26 2009; simplified by Alexander Burstein, Jul 31 2018

From Joerg Arndt, Oct 22 2012: (Start)

Also the number of strings of length 2*n of two different types of balanced parentheses.

For example, a(1) = 2, since the two possible strings of length 2 are [] and (), a(2) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][].

The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)

Number of Dyck paths of length 2n in which the step U=(1,1) come in 2 colors. - José Luis Ramírez Ramírez, Jan 31 2013

Row sums of triangle in A085880. - Philippe Deléham, Nov 15 2013

Hankel transform is 2^(n+n^2) = A053763(n+1). - Philippe Deléham, Nov 15 2013

LINKS

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

Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

J. Bouttier, P. Di Francesco and E. Guitter, Statistics of planar graphs viewed from a vertex: a study via labeled trees, Nucl. Phys. B, Vol. 675, No. 3 (2003), pp. 631-660. See p. 631, eq. (3.3).

Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont and Światosław R. Gal, Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, arXiv:2104.14530 [math.RT], 2021.

Vedran Čačić and Vjekoslav Kovač, On the fraction of IL formulas that have normal forms, arXiv:1309.3408 [math.LO], 2013.

Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.5.

Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schröder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=b=2.

Nicolas Crampe, Julien Gaboriaud and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.

Hsien-Kuei Hwang, Mihyun Kang and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.

Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.

Georg Muntingh, Implicit Divided Differences, Little Schröder Numbers and Catalan Numbers, J. Int. Seq., Vol. 15 (2012), Article 12.6.5; arXiv preprint, arXiv:1204.2709 [math.CO], 2012.

FORMULA

a(n) = 2^n * A000108(n). - Philippe Deléham, Feb 01 2009

From Gary W. Adamson, Jul 12 2011: (Start)

a(n) is the top left term in M^n, M = the following infinite square production matrix:

   2, 2, 0, 0, 0, 0, ...

   2, 2, 2, 0, 0, 0, ...

   2, 2, 2, 2, 0, 0, ...

   2, 2, 2, 2, 2, 0, ...

   2, 2, 2, 2, 2, 2, ...

   ...

(End)

E.g.f.: KummerM(1/2, 2, 8*x). - Peter Luschny, Aug 26 2012

E.g.f.: Let F(x)=Sum_{n>=0} a(n)*x^n/(2*n)!, then F(x) = E(0)/(1-sqrt(x)) where E(k) = 1 - sqrt(x)/(1 - sqrt(x)/(sqrt(x) - (k+1)*(k+2)/2/E(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013

G.f.: 1 + 4*x/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013

G.f.: sqrt(2-8*x-2*sqrt(1-8*x))/(4*x). - Mark van Hoeij, May 10 2013

G.f.: (1-sqrt(1-8*x))/(4*x). - Philippe Deléham, Nov 15 2013

D-finite with recurrence (n+1)*a(n) + 4*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Mar 05 2014

a(n) = 4^n*2F1((1-n)/2,-n/2;1;1)/(n+1). - Benedict W. J. Irwin, Jul 12 2016

a(n) ~ 8^n*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016

From Peter Bala, Aug 17 2021: (Start)

a(n) = Sum_{k = 0..floor(n/2)} A046521(n,2*k)*Catalan(2*k).

G.f.: A(x) = 1/sqrt(1 - 4*x)*e(x/(1 - 4*x)), where e(x) = (c(x) + c(-x))/2 is the even part of the function c(x) = (1 - sqrt(1 - 4*x))/(2*x), the g.f. of the Catalan numbers A000108. Inversely, (c(x) + c(-x))/2 = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).

x*A(x) = Series reversion of (x - 2*x^2). (End)

Sum_{n>=0} 1/a(n) = 68/49 + 96*arctan(1/sqrt(7)) / (49*sqrt(7)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 20/27 - 16*log(2)/81. - Amiram Eldar, Jan 25 2022

MAPLE

A151374_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

for w from 1 to n do a[w] := 2*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od;

convert(a, list)end: A151374_list(23); # Peter Luschny, May 19 2011

MATHEMATICA

aux[i_Integer, j_Integer, n_Integer] := 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, 1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

PROG

(MAGMA) [2^n * Catalan(n): n in [0..25]]; // Vincenzo Librandi, Oct 24 2012

(PARI) x='x+O('x^66); Vec(sqrt(2-8*x-2*sqrt(1-8*x))/(4*x)) \\ Joerg Arndt, May 11 2013

(Sage)

def A151374():

    a, n = 1, 1

    while True:

        yield a

        n += 1

        a = a * (8*n - 12) // n

A = A151374()

print([next(A) for _ in range(24)]) # Peter Luschny, Nov 30 2016

CROSSREFS

Cf. A000108, A046521, A052701, A053763, A085880.

Sequence in context: A074601 A214760 A052701 * A177408 A289431 A337912

Adjacent sequences:  A151371 A151372 A151373 * A151375 A151376 A151377

KEYWORD

nonn,walk

AUTHOR

Manuel Kauers, Nov 18 2008

STATUS

approved

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Last modified June 30 12:16 EDT 2022. Contains 354939 sequences. (Running on oeis4.)