A219671 -id:A219671 - cp's OEIS frontend

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.

0, 1, 18, 294, 5776, 117045, 2505006, 55138293, 1245056184, 28643604147, 669304345150, 15838583011812, 378828554265096, 9143273873757283, 222407411228180010, 5446827816890184990, 134191612737844924608, 3323506599627088488579, 82700482246125321972582

Offset: 0

Jon Perry, Nov 24 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A219671, A219986.

Cf. A002426, A005717.

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],
[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],
[-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]];
function inc(arr,m) {
al=arr.length-1;
full=true;
for (ac=0;ac<=al;ac++) if (arr[ac]!=m) {full=false;break;}
if (full==true) return false;
while (arr[al]==m && al>0) {arr[al]=0;al--;}
arr[al]++;
return true;
}
for (k=0;k<6;k++) {
c=0;
a=new Array();
for (i=0;i

a:= proc(n) a(n):= `if`(n<6, [0, 1, 18, 294, 5776, 117045][n+1],
     (n*(n-1)*(453658*n^4-2664929*n^3+6608535*n^2-8353208*n+3876664)
      *a(n-1) +3*(n-1)*(286527*n^5+2962040*n^4-19850405*n^3+25517846
      *n^2+20905560*n-41336424) *a(n-2) -18*(n-2)*(2*n-5)*(1294945*n^4
      -12949450*n^3+54428897*n^2-110276360*n+88672932) *a(n-3) -81*(n-3)
      *(286527*n^5-10125215*n^4+111022145*n^3-530226521*n^2+1163720520*n
      -966508776) *a(n-4) +729*(453658*n^4-6408231*n^3+34683300*n^2
      -84691467*n+77744124)*(n-4)^2 *a(n-5) +19683*(-42552+15593*n)
      *(n-4)^2 *(n-5)^3 *a(n-6))/ (n^2*(n+1)*(n-1)^2*(15593*n-35413)))
    end:
seq (a(n), n=0..30);  # Alois P. Heinz, Nov 28 2012
A005717 := n -> simplify(GegenbauerC(n-1,-n,-1/2));
A002426 := n -> simplify(GegenbauerC(n,-n,-1/2));
seq( A002426(n)^2 * A005717(n), n=0..30 );  # Mark van Hoeij, Nov 13 2022

a(n) ~ 3^(3*n+3/2) / (4*Pi*n)^(3/2). - Vaclav Kotesovec, Sep 07 2014
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
a(n) = A002426(n)^2 * A005717(n). - Mark van Hoeij, Nov 13 2022

More terms from Alois P. Heinz, Nov 28 2012

A219986 Number of n-step paths on a quartic lattice that move from (0,0,0,0) to (1,0,0,1) allowing all moves in {-1,0,1}^4 inclusive the zero move.

Original entry on oeis.org

0, 1, 36, 1764, 92416, 5267025, 315630756, 19684370601, 1264977082944, 83266957255329, 5588691282002500, 381203015928291216, 26357375491548319296, 1843677173726039815969, 130261796682232750056900, 9284172482167489217304900, 666818520819487582805692416

Offset: 0

Jon Perry, Dec 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A219670, A219671.

b=new Array();
for (i1=-1;i1<2;i1++)
for (i2=-1;i2<2;i2++)
for (i3=-1;i3<2;i3++)
for (i4=-1;i4<2;i4++) {
n=(i1+1)+(i2+1)*3+(i3+1)*9+(i4+1)*27;
b[n]=[i1,i2,i3,i4];
}
function inc(arr,m) {
al=arr.length-1;
full=true;
for (ac=0;ac<=al;ac++) if (arr[ac]!=m) {full=false;break;}
if (full==true) return false;
while (arr[al]==m && al>0) {arr[al]=0;al--;}
arr[al]++;
return true;
}
for (k=0;k<5;k++) {
c=0;
a=new Array();
for (i=0;i

b:= proc(n, p) option remember; `if`(p[4]>n, 0, `if`(n=0, 1,
      add(add(add(add(b(n-1, sort(map(abs, p+[i, j, k, m])))
      , i=-1..1), j=-1..1), k=-1..1), m=-1..1)))
    end:
a:= n-> b(n, [0$2, 1$2]):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2012

b[n_, p_] := b[n, p] = If[p[[4]] > n, 0, If[n == 0, 1,
     Sum[Sum[Sum[Sum[b[n-1, Sort[Abs[ p + {i, j, k, m}]]],
     {i, -1, 1}], {j, -1, 1}], {k, -1, 1}], {m, -1, 1}]]];
a[n_] := b[n, {0, 0, 1, 1}];
Table [a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 02 2012

cp's OEIS Frontend

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.

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