1,2

Number of planar lattice walks of length 3n starting and ending at (0,0), remaining in the first quadrant and using only NE,W,S steps.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Kreweras, Sur une classe de problemes de denombrement lies au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle, Institut de Statistique, Universit\'{e} de Paris, 6 (1965), circa p. 82.

M. Bousquet-M\'elou, Walks in the quarter plane: Kreweras' algebraic model

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

G.f.: (1/(12*x)) * (hypergeom([ -2/3, -1/3],[1/2],27*x)-1) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 02 2009]

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, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] - Manuel Kauers (manuel(AT)kauers.de), Nov 18 2008

(PARI) a(n)=if(n<0, 0, 4^n*(3*n)!/(n+1)!/(2*n+1)!)

Equals 2^(n-1) * A000309(n-1) for n>1.

Cf. A098272. First row of array A098273.

Sequence in context: A011553 A123898 A118644 this_sequence A051711 A012683 A012677

Adjacent sequences: A006332 A006333 A006334 this_sequence A006336 A006337 A006338

nonn,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 20 2008 at the suggestion of R. J. Mathar