A151483 - OEIS

%I #22 Dec 04 2016 13:57:04

%S 1,1,4,12,48,192,832,3712,17152,81152,392192,1928192,9621504,48623616,

%T 248463360,1282031616,6672285696,34993274880,184793432064,

%U 981947645952,5247335399424,28185150357504,152104870084608,824404913160192,4486067252101120,24501262150008832,134274187559698432,738200201575006208

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

%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, 2008.

%F G.f.: ((1-6*x)*(1-4*x-12*x^2)^(1/2)-4*x^2+8*x-1)/(32*x^3). - _Mark van Hoeij_, Aug 20 2014

%F a(n) = sqrt(-1/3)*(-2)^n*hypergeom([1/2, n+4],[2],4/3)/(n+1). - _Mark van Hoeij_, Aug 23 2014

%F Conjecture: +(n+3)*a(n) -4*n*a(n-1) +12*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016

%p A151483 := proc(n)

%p   coeftayl(((1-6*x)*(1-4*x-12*x^2)^(1/2)-4*x^2+8*x-1)/(32*x^3), x=0, n);

%p end proc:

%p seq(A151483(n), n=0..30); # _Wesley Ivan Hurt_, Aug 23 2014

%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, 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[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

%t CoefficientList[Series[((1 - 6x)(1 - 4x - 12x^2)^(1/2) - 4x^2 + 8x - 1)/(32 x^3), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Aug 23 2014 *)

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008