A151464 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, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, 0)}.

1, 1, 4, 12, 46, 180, 745, 3185, 14000, 62832, 287154, 1331484, 6251916, 29671356, 142132848, 686420592, 3338939032, 16345771728, 80480627656, 398307700648, 1980504505408, 9889617286848, 49575852422122, 249406833948012, 1258841279547604, 6373077654620340, 32355358786303440, 164693131263424560

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..462

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

Formula

G.f.: ((1+1/x)*Int(((8*x^2+4*x+1)*hypergeom([1/4, 3/4],[1],64*x^3*(2*x+1)/(8*x^2-1)^2)-12*x^2*(2*x+1)*hypergeom([1/4, 3/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2))/((x+1)^2*(1-8*x^2)^(3/2)),x)-1)/(2*x). - Mark van Hoeij, Aug 20 2014

Maple

M[0]:= Matrix(1, 1, 1):
for i from 1 to 100 do
  M[i]:= Matrix(i+1, i+1);
  for p in [[-1, -1], [-1, 0], [-1, 1], [0, -1], [0, 1], [1, 0]] do
    j1:= max(1, 1+p[1]); j2:= max(1, 1+p[2]);
    if j1 <= p[1]+i and j2 <= p[2]+i then
      M[i][j1..p[1]+i, j2..p[2]+i] := M[i][j1..p[1]+i, j2..p[2]+i]
            + M[i-1][j1-p[1]..i, j2-p[2]..i]
    fi
  od
od:
seq(add(M[i][1, j], j=1..i+1), i=0..100); # Robert Israel, Aug 20 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[i, -1 + j, -1 + n] + aux[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[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A126202 A149373 A259203 * A101225 A000775 A149374

Adjacent sequences: A151461 A151462 A151463 * A151465 A151466 A151467

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved