A151483 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0)}.

1, 1, 4, 12, 48, 192, 832, 3712, 17152, 81152, 392192, 1928192, 9621504, 48623616, 248463360, 1282031616, 6672285696, 34993274880, 184793432064, 981947645952, 5247335399424, 28185150357504, 152104870084608, 824404913160192, 4486067252101120, 24501262150008832, 134274187559698432, 738200201575006208

Offset

0,3

Links

Table of n, a(n) for n=0..27.

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

Formula

G.f.: ((1-6*x)*(1-4*x-12*x^2)^(1/2)-4*x^2+8*x-1)/(32*x^3). - Mark van Hoeij, Aug 20 2014

a(n) = sqrt(-1/3)*(-2)^n*hypergeom([1/2, n+4],[2],4/3)/(n+1). - Mark van Hoeij, Aug 23 2014

Conjecture: +(n+3)*a(n) -4*n*a(n-1) +12*(-n+1)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

Maple

A151483 := proc(n)
  coeftayl(((1-6*x)*(1-4*x-12*x^2)^(1/2)-4*x^2+8*x-1)/(32*x^3), x=0, n);
end proc:
seq(A151483(n), n=0..30); # Wesley Ivan Hurt, Aug 23 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, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
CoefficientList[Series[((1 - 6x)(1 - 4x - 12x^2)^(1/2) - 4x^2 + 8x - 1)/(32 x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 23 2014 *)

Crossrefs

Sequence in context: A149385 A092898 A110594 * A355800 A192622 A324801

Adjacent sequences: A151480 A151481 A151482 * A151484 A151485 A151486

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved