A151472 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), (0, -1), (0, 1), (1, -1), (1, 0)}.

1, 1, 3, 9, 30, 110, 423, 1687, 6984, 29574, 128074, 564652, 2527292, 11463972, 52602015, 243824807, 1140448152, 5377337150, 25539196048, 122093592944, 587170555168, 2839207157456, 13797304069674, 67357039620092, 330225541717108, 1625329978935340, 8028874036140468, 39796190100237612

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 [math.CO], 2008.

Formula

G.f.: Int(Int(Int(x*(-36-2*Int((1-4*x-12*x^2)^(3/2)*((256*x^5+416*x^4+128*x^3+3*x+3)*hypergeom([5/4, 7/4],[1],64*x^3*(2*x+1)/(8*x^2-1)^2)-7*x*(40*x^4+68*x^3-4*x^2-18*x-3)*hypergeom([5/4, 11/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2))/((2*x+1)*(1-8*x^2)^(7/2)*x^2),x))/(1-4*x-12*x^2)^(5/2),x),x),x)/x^3. - Mark van Hoeij, Aug 27 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, 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: A246472 A091699 A129167 * A107379 A117428 A339835

Adjacent sequences: A151469 A151470 A151471 * A151473 A151474 A151475

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved