A151366 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0)}.

1, 0, 2, 2, 12, 30, 130, 462, 1946, 7980, 34776, 153120, 694056, 3194334, 14971242, 71133062, 342500730, 1667918824, 8208038124, 40772105244, 204270936480, 1031413134960, 5245260798960, 26850869456400, 138289429433200, 716247599547360, 3729128330979200, 19510354349803200, 102540704879774160

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..500

Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.

Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.

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

Formula

G.f.: ((2*x+1)*(hypergeom([-2/3, -1/3],[1],27*x^2*(2*x+1))+4*x*hypergeom([-1/3, 1/3],[2],27*x^2*(2*x+1)))/(3*x+1)-1-3*x-5*x^2)/(3*x^3). - Mark van Hoeij, Aug 17 2014

Conjecture: (n+4)*(n+3)*a(n) -n*(n-1)*a(n-1) -12*(2*n+1)*(n-1)*a(n-2) -36*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 21 2017

G.f.: (-5*x^2-3*x-1)/(3*x^3)+(1+3*x)^3*hypergeom([1/4, 3/4],[2],64*x*(1+3*x)^3/(1+24*x+36*x^2)^2)/(3*x^3*(1+24*x+36*x^2)^(1/2)). - Mark van Hoeij, Jul 27 2021

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[aux[0, 0, n], {n, 0, 25}]

Crossrefs

Sequence in context: A130306 A199127 A093044 * A184944 A033886 A185144

Adjacent sequences: A151363 A151364 A151365 * A151367 A151368 A151369

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved