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%I #16 Apr 09 2018 22:32:28
%S 1,0,1,2,4,8,23,58,142,352,982,2620,6910,18328,51361,141022,385266,
%T 1055936,2981288,8329488,23219142,64864824,184401956,521245736,
%U 1471964582,4163298128,11904910448,33933381248,96692013158,275858766248,792661868663,2273584990178,6521455463978,18724412753408
%N 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), (0, -1), (1, 1)}.
%H Robert Israel, <a href="/A151380/b151380.txt">Table of n, a(n) for n = 0..688</a>
%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008.
%F G.f.: Int((6*x^3+2*x^2+x-1)*Int(x*(16*x^3+6*x^2-6*x+1+2*x^2*(1-17*x^2+6*x^3+22*x^4)*hypergeom([3/4, 5/4],[2],64*x^4)-(44*x^4-16*x^2+6*x-1)*hypergeom([1/4, 3/4],[1],64*x^4))/((6*x^3+2*x^2+x-1)^2*(1-2*x-3*x^2)^(1/2)),x)/((1-2*x-3*x^2)^(1/2)*x^2),x)/(x*(x-1)). - _Mark van Hoeij_, Aug 25 2014
%p f:= proc(a, b, n) option remember;
%p local t;
%p if n = 0 then if a = 0 then return 1 else return 0 fi fi;
%p if a > n then return 0 fi;
%p t:= procname(a+1, b+1, n-1);
%p if a >= 1 then t:= t + procname(a-1,b+1,n-1) fi;
%p if b >= 1 then t:= t + procname(a,b-1,n-1) fi;
%p t;
%p end proc:
%p seq(f(0, 0, n), n=0..50); # _Robert Israel_, Apr 09 2018
%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[1 + i, -1 + j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]
%K nonn,walk
%O 0,4
%A _Manuel Kauers_, Nov 18 2008
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