A151410 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, 0), (1, -1), (1, 0), (1, 1)}.

1, 2, 10, 65, 490, 4032, 35244, 321750, 3035890, 29395652, 290621188, 2922898706, 29821640380, 307994453600, 3214454901480, 33855533036865, 359438259174930, 3843173300937300, 41351489731559700, 447450028715934090, 4866409456815200580, 53171146669028038560, 583400942149413843000

Offset

0,2

Links

Table of n, a(n) for n=0..22.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Formula

G.f.: Int(hypergeom([1/2,3/2],[2],16*x/(1+4*x))/(1+4*x)^(1/2),x)/x. - Mark van Hoeij, Aug 20 2014

a(n) = (-1)^n*(1+n/3)*binomial(2*n+1,n)*hypergeom([5/2,-n],[4],4)/(2*n+1) = A000108(n)*(A001006(n)+A001006(n+1))*(n+3)/6. - Mark van Hoeij, Aug 24 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, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

Crossrefs

Sequence in context: A130721 A167449 A064170 * A230050 A340467 A278459

Adjacent sequences: A151407 A151408 A151409 * A151411 A151412 A151413

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved