A151437 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, 1), (0, -1), (1, -1), (1, 1)}.

1, 0, 2, 2, 14, 28, 148, 408, 1948, 6316, 29164, 103336, 473860, 1770364, 8141872, 31491484, 145696432, 577601620, 2689508560, 10864037780, 50889023352, 208655144644, 982532205120, 4078460733332, 19292955235136, 80919972951468, 384307594445168, 1626307851201068, 7750396627918568

Offset

0,3

Links

Table of n, a(n) for n=0..28.

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((450*x^3+90*x^2-42*x-2)*(1+3*Int((1-2*x-15*x^2)^(1/2)*(x-1+((16*x^2+1)*(19872*x^6+3753*x^5-5565*x^4-510*x^3+294*x^2+20*x-8)*hypergeom([5/4, 7/4],[2],64*x^2*(1+x^2)/(16*x^2+1)^2)-10*x^2*(2880*x^6-2205*x^5-2139*x^4+423*x^3+45*x^2+12*x-8)*hypergeom([7/4, 9/4],[3],64*x^2*(1+x^2)/(16*x^2+1)^2))/(16*x^2+1)^(7/2))/ (225*x^3+45*x^2-21*x-1)^2,x))/(1-2*x-15*x^2)^(3/2),x),x)/(x^2*(x-1)). - 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, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + 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: A032038 A187734 A151353 * A235349 A226157 A264508

Adjacent sequences: A151434 A151435 A151436 * A151438 A151439 A151440

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved