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A219670 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction on {-1,0,1}^3 and zero moves allowed.

%I A219670 #36 Nov 14 2022 02:14:13
%S A219670 0,1,18,294,5776,117045,2505006,55138293,1245056184,28643604147,
%T A219670 669304345150,15838583011812,378828554265096,9143273873757283,
%U A219670 222407411228180010,5446827816890184990,134191612737844924608,3323506599627088488579,82700482246125321972582
%N A219670 Number of n-step paths on cubic lattice from (0,0,0) to (1,0,0) with moves in any direction on {-1,0,1}^3 and zero moves allowed.
%H A219670 Alois P. Heinz, <a href="/A219670/b219670.txt">Table of n, a(n) for n = 0..250</a>
%F A219670 a(n) ~ 3^(3*n+3/2) / (4*Pi*n)^(3/2). - _Vaclav Kotesovec_, Sep 07 2014
%F A219670 Recurrence (of order 4): (n-1)^2*n^2*(n+1)*(2*n-5)*(7*n^4 - 56*n^3 + 166*n^2 - 216*n + 105)*a(n) = (n-1)*n*(2*n-5)*(2*n-1)*(70*n^6 - 630*n^5 + 2206*n^4 - 3759*n^3 + 3181*n^2 - 1188*n + 144)*a(n-1) + 3*(n-1)*(2*n-3)*(490*n^8 - 5880*n^7 + 30030*n^6 - 85050*n^5 + 145359*n^4 - 152064*n^3 + 93599*n^2 - 30264*n + 3852)*a(n-2) - 27*(n-2)^2*(2*n-5)*(2*n-1)*(70*n^6 - 630*n^5 + 2206*n^4 - 3813*n^3 + 3424*n^2 - 1563*n + 342)*a(n-3) - 729*(n-3)^3*(n-2)^2*(2*n-1)*(7*n^4 - 28*n^3 + 40*n^2 - 24*n + 6)*a(n-4). - _Vaclav Kotesovec_, Sep 07 2014
%F A219670 a(n) = A002426(n)^2 * A005717(n). - _Mark van Hoeij_, Nov 13 2022
%p A219670 a:= proc(n) a(n):= `if`(n<6, [0, 1, 18, 294, 5776, 117045][n+1],
%p A219670      (n*(n-1)*(453658*n^4-2664929*n^3+6608535*n^2-8353208*n+3876664)
%p A219670       *a(n-1) +3*(n-1)*(286527*n^5+2962040*n^4-19850405*n^3+25517846
%p A219670       *n^2+20905560*n-41336424) *a(n-2) -18*(n-2)*(2*n-5)*(1294945*n^4
%p A219670       -12949450*n^3+54428897*n^2-110276360*n+88672932) *a(n-3) -81*(n-3)
%p A219670       *(286527*n^5-10125215*n^4+111022145*n^3-530226521*n^2+1163720520*n
%p A219670       -966508776) *a(n-4) +729*(453658*n^4-6408231*n^3+34683300*n^2
%p A219670       -84691467*n+77744124)*(n-4)^2 *a(n-5) +19683*(-42552+15593*n)
%p A219670       *(n-4)^2 *(n-5)^3 *a(n-6))/ (n^2*(n+1)*(n-1)^2*(15593*n-35413)))
%p A219670     end:
%p A219670 seq (a(n), n=0..30);  # _Alois P. Heinz_, Nov 28 2012
%p A219670 A005717 := n -> simplify(GegenbauerC(n-1,-n,-1/2));
%p A219670 A002426 := n -> simplify(GegenbauerC(n,-n,-1/2));
%p A219670 seq( A002426(n)^2 * A005717(n), n=0..30 );  # _Mark van Hoeij_, Nov 13 2022
%o A219670 (JavaScript)
%o A219670 b=[[1,1,1],[1,1,0],[1,1,-1],[1,0,1],[1,0,0],[1,0,-1],[1,-1,1],[1,-1,0],[1,-1,-1],
%o A219670 [0,1,1],[0,1,0],[0,1,-1],[0,0,1],[0,0,0],[0,0,-1],[0,-1,1],[0,-1,0],[0,-1,-1],
%o A219670 [-1,1,1],[-1,1,0],[-1,1,-1],[-1,0,1],[-1,0,0],[-1,0,-1],[-1,-1,1],[-1,-1,0],[-1,-1,-1]];
%o A219670 function inc(arr,m) {
%o A219670 al=arr.length-1;
%o A219670 full=true;
%o A219670 for (ac=0;ac<=al;ac++) if (arr[ac]!=m) {full=false;break;}
%o A219670 if (full==true) return false;
%o A219670 while (arr[al]==m && al>0) {arr[al]=0;al--;}
%o A219670 arr[al]++;
%o A219670 return true;
%o A219670 }
%o A219670 for (k=0;k<6;k++) {
%o A219670 c=0;
%o A219670 a=new Array();
%o A219670 for (i=0;i<k;i++) a[i]=0;
%o A219670 for (i=0;i<Math.pow(27,k);i++) {
%o A219670 p=[0,0,0];
%o A219670 for (j=0;j<k;j++) {p[0]+=b[a[j]][0];p[1]+=b[a[j]][1];p[2]+=b[a[j]][2];}
%o A219670 if (p[0]==1 && p[1]==0 && p[2]==0) c++;
%o A219670 inc(a,26);
%o A219670 }
%o A219670 document.write(c+", ");
%o A219670 }
%Y A219670 Cf. A219671, A219986.
%Y A219670 Cf. A002426, A005717.
%K A219670 nonn
%O A219670 0,3
%A A219670 _Jon Perry_, Nov 24 2012
%E A219670 More terms from _Alois P. Heinz_, Nov 28 2012