A151362 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2*n steps taken from {(-1, -1), (-1, 0), (-1, 1), (1, -1), (1, 0), (1, 1)}.

1, 2, 18, 255, 4522, 91896, 2047452, 48748986, 1220457810, 31779889284, 854110511124, 23559266827278, 664125694509564, 19070108145820400, 556345776173277960, 16455889048642607295, 492658546882981692690, 14907686709710614053300, 455413194094843994648100

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..583

Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner, Composition schemes: q-enumerations and phase transitions in Gibbs models, 2024. See p. 8.

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

Formula

a(n) = hypergeom([-n, 1/2-n], [2], 4)*binomial(2*n, n)/(n+1). - Robert Israel, Aug 14 2014

a(n) = M(2n)*C(n), where M(n) denotes Motzkin numbers, A001006, and C(n) the Catalan numbers A000108. Proof via a bijection of a pair of Dyck excursion and Motzkin excursion. - Markus Kuba, May 05 2022

Maple

seq(add(binomial(2*n, 2*k)*binomial(2*k, k)/(k+1), k=0..n)*binomial(2*n, n)/(n+1), n=0..18);  # Mark van Hoeij, May 12 2013
S := proc(a) global x; series(a, x=0, 20) end:
ogf := S(int(x^(-1/2)*int(S(x^(-1/2)*hypergeom([3/4, 5/4], [2], 64*x/(12*x+1)^2)/(12*x+1)^(3/2)), x), x)/(2*x)); #  Mark van Hoeij, Aug 14 2014
# third Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
     (4*n*(4*n-1)*(2*n-1)*(10*n^2-5*n-3) *a(n-1)
      -36*(n-1)*(2*n-1)*(4*n+1)*(-3+2*n)^2 *a(n-2))/
      (n*(1+2*n)*(4*n-3)*(n+1)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 14 2014

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

Crossrefs

Cf. A000108, A001006.

Sequence in context: A109517 A213643 A143138 * A215362 A360974 A350461

Adjacent sequences: A151359 A151360 A151361 * A151363 A151364 A151365

Keyword

nonn,walk,changed

Author

Manuel Kauers, Nov 18 2008

Status

approved