A065078 Triangle read by rows: a(n,m) = T[n,m,m] where T[i,j,k] is the 3-dimensional pyramid defined by T[n,m,0]=1 and T[i,j,k]=0 if j>i or k>j and T[i,j,k]=T[i-1,j,k]+T[i,j-1,k]+T[i,j,k-1].

1, 1, 1, 1, 2, 3, 1, 3, 10, 18, 1, 4, 22, 79, 162, 1, 5, 40, 220, 831, 1851, 1, 6, 65, 492, 2681, 10488, 24661, 1, 7, 98, 962, 6883, 37367, 149743, 365613, 1, 8, 140, 1715, 15318, 105731, 573051, 2336243, 5863881, 1, 9, 192, 2856, 30840, 258604, 1742770

Offset

0,5

Comments

Number of paths to T[n,m,m] counted from the bottom plane (or T[n,m,0]).

Links

Table of n, a(n) for n=0..51.

Wouter Meeussen, Further comments on this sequence

Formula

T[0, 0, 0] := 1; T[x_, y_, z_] := 0 /; (x< y || y< z); T[u_, v_, 0] := 1; T[_, 0, 0] := 1; T[x_, y_, z_] := (T[x, y, z]= T[x-1, y, z]+T[x, y-1, z] +T[x, y, z-1]) /; (y<=x ||z<=y); Table[T[x, y, y], {x, 0, 10}, {y, 0, x}]

Example

[3,2,2] can be reached from 3*[1,1,0] + 3*[2,1,0] + 2*[2,2,0] + 1*[3,1,0] + 1*[3,2,0], so a(3,2) = 3 + 3 + 2 + 1 + 1 =10.
Triangle begins
1;
1, 1;
1, 2, 3;
1, 3, 10, 18;
1, 4, 22, 79, 162;

Crossrefs

Last number in each row is A065058.

Cf. A005789, A065057.

Sequence in context: A151880 A108990 A145080 * A203989 A126744 A126736

Adjacent sequences: A065075 A065076 A065077 * A065079 A065080 A065081

Keyword

nonn,tabl

Author

Wouter Meeussen, Nov 09 2001

Status

approved