A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

1, 1, 2, 2, 1, 4, 4, 4, 8, 4, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32

Offset

0,3

Comments

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

The sum of the numbers in each slice of the pyramid is 5^n.

The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each antidiagonal of the j-th triangle of the n-th slice of A261358. - Dimitri Boscainos, Aug 21 2015

Links

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

Formula

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

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

Example

Here is the fourth (n=3) slice of the pyramid:
        1
      6   6
   12  24  12
  8  24  24   8

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) +2*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);
# Adapted from Alois P. Heinz's Maple program for A261356

Prog

(PARI) tabf(nn) = {for (n=0, nn, for (j=0, n, for (k=0, j, print1(2^j*binomial(n, j)*binomial(j, k), ", ")); print(); ); print(); ); } \\ Michel Marcus, Oct 07 2015

Crossrefs

Cf. A046816, A261356, A261358.

Sequence in context: A136787 A165038 A305191 * A238870 A213946 A145036

Adjacent sequences: A261354 A261355 A261356 * A261358 A261359 A261360

Keyword

nonn,tabf,walk

Author

Dimitri Boscainos, Aug 16 2015

Status

approved