A151403 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2*n steps taken from {(-1, 0), (-1, 1), (1, 0), (1, 1)}.

1, 4, 32, 320, 3584, 43008, 540672, 7028736, 93716480, 1274544128, 17611882496, 246566354944, 3489862254592, 49855175065600, 717914520944640, 10409760553697280, 151860036312760320, 2227280532587151360, 32823081532863283200, 485781606686376591360, 7217326727911880785920

Offset

0,2

Comments

Essentially the same as A052704. - R. J. Mathar, Nov 27 2008

From Joerg Arndt, Oct 22 2012: (Start)

Number of strings of length 2*n of four different types of balanced parentheses.

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) comes in 4 colors. - José Luis Ramírez Ramírez, Jan 31 2013

References

Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 106.

Links

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

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

Alon Regev, Enumerating Triangulations by Parallel Diagonals, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.5; arXiv preprint, arXiv:1208.3915 [math.CO], 2012.

Formula

a(n) = 4^n*A000108(n). - Philippe Deléham, Feb 01 2009

a(n) = Integral_{x=-2..2} (2*x)^(2*n)*sqrt((2-x)*(2+x)))/(2*Pi) dx. - Peter Luschny, Sep 11 2011

E.g.f.: KummerM(1/2, 2, 16*x). - Peter Luschny, Aug 26 2012

G.f.: 2/(1 + sqrt(1-16*x)) = 1/U(0) where U(k) = 1 - 4*x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 30 2012

G.f.: c(4*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013

a(n) = sum{k=0..n} A085880(n,k)*3^k. - Philippe Deléham, Nov 15 2013

a(n) = 4^n*hypergeom([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014

a(n) = 4^n*(2*n)!*[x^(2*n)]hypergeom([],[2],x^2). - Peter Luschny, Jan 31 2015

a(n) ~ 2^(4*n+2)/((2*n+1)*sqrt(Pi*(4*n+5))). - Peter Luschny, Jan 31 2015

D-finite with recurrence: (n+1)*a(n) +8*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

Sum_{n>=0} 1/a(n) = 88/75 + 128*arctan(1/sqrt(15)) / (75*sqrt(15)). - Vaclav Kotesovec, Nov 23 2021

Sum_{n>=0} (-1)^n/a(n) = 248/289 - 384*arctanh(1/sqrt(17)) / (289*sqrt(17)). - Amiram Eldar, Jan 25 2022

G.f.: 1/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-4*x/(1-...)))))))))(continued fraction), cf. g.f. by Sergei N. Gladkovskii. - Nikolaos Pantelidis, Nov 21 2022

Maple

A151403_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 4*(a[w-1]+add(a[j]*a[w-j-1], j=1..w-1)) od; convert(a, list) end: A151403_list(20); # Peter Luschny, May 19 2011
seq(4^n*(2*n)!*coeff(series(hypergeom([], [2], x^2), x, 2*n+2), x, 2*n), n=0..20); # Peter Luschny, Jan 31 2015

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] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

Prog

(Magma) [4^n * Catalan(n): n in [0..20]]; // Vincenzo Librandi, Oct 24 2012
(Sage)
A151403 = lambda n: 4^n*hypergeometric([1-n, -n], [2], 1)
[Integer(A151403(n).n()) for n in range(21)] # Peter Luschny, Sep 22 2014

Crossrefs

Cf. A000108, A052704, A085880.

Sequence in context: A220118 A060174 A052704 * A289427 A090004 A357404

Adjacent sequences: A151400 A151401 A151402 * A151404 A151405 A151406

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved