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%I #56 Jan 12 2023 02:28:36
%S 1,9,61,377,2241,13073,75517,433905,2485825,14218905,81270333,
%T 464387817,2653649025,15167050785,86716873725,495998874593,
%U 2838240338817,16248650965289,93065296937533,533285164334169,3057236753252161,17534423944871729,100609937775369981
%N Number of walks from (0,0) to (n+3,n) which take steps from {E, N, NE}.
%C +-3-diagonal of A008288 as a square array. - _Shel Kaphan_, Jan 07 2023
%D S. Gao, H. Niederhausen, Counting New Lattice Paths and Walks with Several Step Vectors (submitted to Congr. Numer.). - _Shanzhen Gao_, May 25 2011
%H Alois P. Heinz, <a href="/A190666/b190666.txt">Table of n, a(n) for n = 0..400</a>
%H Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2749.pdf">Counting Lattice Paths and Walks with Several Step Vectors</a>, FCS 2014.
%F a(n) = Sum_{k=0..n} C(n,k) * C(n+k+3,k+3) = A113139 (n+3,3). - _Alois P. Heinz_, Jun 01 2011
%F G.f.: (-1 + 3*x - x^2 + (1 - 6*x + 6*x^2 - x^3)/sqrt(x^2 - 6*x + 1))/(2*x^3). - _Alois P. Heinz_, Jun 03 2011
%F Recurrence: n*(n+3)*a(n) = (5*n^2 + 15*n + 16)*a(n-1) + (5*n^2 - 5*n + 6)*a(n-2) - (n-2)*(n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ sqrt(1632 + 1154*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 20 2012
%F From _Peter Bala_, Mar 02 2017: (Start)
%F a(n) = (1/2^(n+1))*Sum_{k >= 3} (1/2^k)*binomial(n+k, k)*binomial(n+k, n+3).
%F a(n) = (-1)^n*Sum_{k = 0..n} (-2)^k*binomial(n,k) * binomial(n+k+3,k).
%F n*(n+3)*(2*n + 1)*a(n) = 6*(n+1)*(2*n^2 + 4*n + 3)*a(n-1) - (n-1)*(n+2)*(2*n + 3)*a(n-2) with a(0) = 1 and a(1) = 9. (End)
%F a(n) = (-1)^n*hypergeom([-n, n+4], [1], 2). - _Peter Luschny_, Mar 02 2017
%p b:= proc(i, j) option remember;
%p if i<0 or j<0 then 0
%p elif i=0 and j=0 then 1
%p else b(i-1, j) +b(i, j-1) +b(i-1, j-1)
%p fi
%p end:
%p a:= n-> b(n+3, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 28 2011
%t b[i_, j_] /; i<0 || j<0 = 0; b[0, 0] = 1; b[i_, j_]:= b[i, j]= b[i-1, j] + b[i, j-1] + b[i-1, j-1]; a[n_] := b[n+3, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 01 2011, after Maple prog. *)
%t CoefficientList[Series[(-1+3*x-x^2+(1-6*x+6*x^2-x^3)/Sqrt[x^2-6*x+1])/(2*x^3), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t Table[(-1)^n Hypergeometric2F1[-n, n+4, 1, 2], {n,0,22}] (* _Peter Luschny_, Mar 02 2017 *)
%Y Cf. A113139, A002002, A026002.
%K nonn,walk,easy
%O 0,2
%A _Shanzhen Gao_, May 25 2011
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