A151367 - OEIS

%I #16 Aug 03 2024 07:12:52

%S 1,0,2,2,13,27,140,392,1882,6289,28906,107949,486438,1948638,8730438,

%T 36611160,164259758,710530289,3203433595,14163150429,64260242637,

%U 288694503092,1318679597635,5996837692998,27572301084897,126595556379751,585652882733959,2709967750078764,12607711205847168

%N 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, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, 1)}.

%H Robert Israel, <a href="/A151367/b151367.txt">Table of n, a(n) for n = 0..700</a>

%H A. Bostan, K. Raschel, and B. Salvy, <a href="http://dx.doi.org/10.1016/j.jcta.2013.09.005">Non-D-finite excursions in the quarter plane</a>, J. Comb. Theory A 121 (2014) 45-63, Table 1 Tag 43, Tag 49.

%H Mireille Bousquet-Mélou and Marni Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%p Steps:= [[-1, -1], [-1, 0], [-1, 1], [0, -1], [0, 1], [1, 1]]:

%p f:= proc(n, p) option remember; local t, s;

%p       if max(p) > n then return 0 fi;

%p       add(procname(n-1, s), s = select(t -> min(t)>=0, map(`+`, Steps, p)))

%p end proc:

%p f(0, [0, 0]):= 1:

%p map(f, [$0..40], [0, 0]); # _Robert Israel_, Aug 02 2024

%t 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[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008