A065058 -id:A065058 - cp's OEIS frontend

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

Wouter Meeussen, Nov 09 2001

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

			[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;

Wouter Meeussen, Further comments on this sequence

Last number in each row is A065058.

Cf. A005789, A065057.

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}]

A065077 Triangle read by rows: T(n,m) = C[n,m,m] where C[i,j,k] is the 3-dimensional Catalan pyramid defined by C[0,0,0]=1 and C[i,j,k]=0 if j>i or k>j and C[i,j,k]=C[i-1,j,k]+C[i,j-1,k]+C[i,j,k-1].

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 1, 6, 21, 42, 1, 10, 56, 210, 462, 1, 15, 120, 660, 2574, 6006, 1, 21, 225, 1650, 9009, 36036, 87516, 1, 28, 385, 3575, 25025, 136136, 554268, 1385670, 1, 36, 616, 7007, 60060, 408408, 2217072, 9145422, 23371634, 1, 45, 936, 12740, 129948

Offset: 0

Wouter Meeussen, Nov 09 2001

T(n,m)= number of standard tableaux of shape (n,m,m) (0Emeric Deutsch, May 14 2004

			1;
1,1;
1,3,5;
1,6,21,42;
1,10,56,210,462;
1,15,120,660,2574,6006;
...
T(2,1)=3 because in the first row of the diagram (2,1,1) we can have 12 or 13 or 14.

Last number in each row is A005789

Cf. A005789, A065058, A065078.

a:=proc(n,m) if m<=n then (n+2*m)!*(n-m+1)*(n-m+2)/m!/(m+1)!/(n+2)! else 0 fi end: seq(seq(a(n,m),m=0..n),n=0..9);

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

T(n, m)=(n+2m)!(n-m+1)(n-m+2)/[m!(m+1)!(n+2)! ] (0<=m<=n). - Emeric Deutsch, May 14 2004

cp's OEIS Frontend

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].

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