OFFSET
0,2
LINKS
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
MathOverFlow, Enumeration of lattice paths of a specific type.
Kilian Raschel, Counting walks in a quadrant: a unified approach via boundary value problems, J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012).
FORMULA
G.f.: (3 + 2*(1 - 4*z^2)*u*Sum_{k>=1}((-1)^k*z^{k-1}*w^k/(1 + z^{k+1}*w^k*u)^2)/(1 - 4*z) where w = (1 - sqrt(1 - 4*z^2))/(2*z^2) = 1 + z^2 + 2*z^4 + 5*z^6 + 14*z^8 + ... and u = ((1 - 3*z)*(1 + z - z^2*w) - sqrt((1 - 2*z)*(1 - 2*z - 7*z^2)*(1 - z^2*w)))/(2*z^3) = 1 + 2*z + 7*z^2 + 20*z^3 + 66*z^4 + ... (see MathOverFlow link). - Mamuka Jibladze, Dec 24 2023
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[1 + i, -1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved