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%I #17 Nov 12 2022 10:48:23
%S 1,3,12,45,180,702,2808,11097,44388,176418,705672,2812482,11249928,
%T 44903484,179613936,717517521,2870070084,11470898106,45883592424,
%U 183438670950,733754683800,2934026948196,11736107792784,46934017407594,187736069630376,750833732416212,3003334929664848
%N Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0)}
%C Hankel transform is 3^C(n+1,2). - _Philippe Deléham_, Feb 01 2009
%C Inverse binomial transform of A151253. - _Philippe Deléham_, Feb 03 2009
%H G. C. Greubel, <a href="/A151162/b151162.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry2/barry73.html">A Note on a One-Parameter Family of Catalan-Like Numbers</a>, JIS 12 (2009) 09.5.4
%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.
%F a(n) = Sum_{k=0..n} A120730(n,k)*3^k. - _Philippe Deléham_, Feb 01 2009
%F From _Philippe Deléham_, Feb 02 2009: (Start)
%F a(2*n+2) = 4*a(2*n+1), a(2*n+1) = 4*a(2*n) - 3^n*A000108(n)
%F a(2*n+1) = 4*a(2*n) - A005159(n).
%F G.f.: (sqrt(1 - 12*x^2) + 6*x - 1)/(6*x*(1 - 4*x)). (End)
%F a(n) ~ 2^(2*n+1)/3. - _Vaclav Kotesovec_, Oct 31 2017
%F a(n) = (4*(n+1)*a(n-1) + 12*(n-2)*a(n-2) - 48*(n-2)*a(n-3))/(n+1). - _G. C. Greubel_, Nov 09 2022
%t aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n]= aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1+n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n,0,30}]
%t a[n_]:= a[n]= If[n<3, (n+2)!/2, (4*(n+1)*a[n-1] +12*(n-2)*a[n-2] -48*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n,0,40}] (* _G. C. Greubel_, Nov 09 2022 *)
%o (Magma) [n le 3 select Factorial(n+1)/2 else (4*n*Self(n-1) + 12*(n-3)*Self(n-2) - 48*(n-3)*Self(n-3))/n: n in [1..41]]; // _G. C. Greubel_, Nov 09 2022
%o (SageMath)
%o def a(n): # a = A151162
%o if (n==0): return 1
%o elif (n%2==1): return 4*a(n-1) - 3^((n-1)/2)*catalan_number((n-1)/2)
%o else: return 4*a(n-1)
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Nov 09 2022
%Y Cf. A000108, A005159, A120730, A151253.
%K nonn,walk
%O 0,2
%A _Manuel Kauers_, Nov 18 2008
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