A261356 Pyramid of coefficients in expansion of (1+x+2*y)^n.

1, 1, 1, 2, 1, 2, 4, 1, 4, 4, 1, 3, 6, 3, 12, 12, 1, 6, 12, 8, 1, 4, 8, 6, 24, 24, 4, 24, 48, 32, 1, 8, 24, 32, 16, 1, 5, 10, 10, 40, 40, 10, 60, 120, 80, 5, 40, 120, 160, 80, 1, 10, 40, 80, 80, 32, 1, 6, 12, 15, 60, 60, 20, 120, 240, 160, 15, 120, 360, 480

Offset: 0

Author

Dimitri Boscainos, Aug 16 2015

Keywords

• nonn
• tabf

Comments

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), (1,1,0) and two kinds of steps (1,1,1).

The sum of the terms in each slice of the pyramid is 4^n (A000302).

The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each row of the j-th triangle of the n-th slice of A189225. - Dimitri Boscainos, Aug 21 2015

Examples

			Here is the fourth (n=3) slice of the pyramid:
.....1......
...3   6....
..3  12  12.
.1  6  12  8
As an irregular triangle, rows begin:
1;
1, 1, 2;
1, 2, 4, 1, 4, 4;
1, 3, 6, 3, 12, 12, 1, 6, 12, 8;
...
		

Links

• Alois P. Heinz, Rows n = 0..38, flattened

Crossrefs

Cf. A046816, A261357.

Programs

Maple

p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Alois P. Heinz, Aug 20 2015

Mathematica

p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + p[i - 1, j, k - 1] + 2* p[i - 1, j - 1, k - 1]]];
Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Sep 16 2023, after Alois P. Heinz *)

Formula

T(i+1,j,k) = 2*T(i,j-1,k-1)+T(i,j-1,k)+T(i,j,k); T(i,j,-1)=0,...; T(0,0,0)=1.

T(n,j,k) = 2^k*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015