A261360 Pentatope of coefficients in expansion of (1 + 2*x + 2*y + 2*z)^n.

1, 1, 2, 2, 2, 1, 4, 4, 4, 4, 8, 8, 4, 8, 4, 1, 6, 6, 6, 12, 24, 24, 12, 24, 12, 8, 24, 24, 24, 48, 24, 8, 24, 24, 8, 1, 8, 8, 8, 24, 48, 48, 24, 48, 24, 32, 96, 96, 96, 192, 96, 32, 96, 96, 32, 16, 64, 64, 96, 192, 96, 64, 192, 192, 64, 16, 64, 96, 64, 96, 1, 10, 10, 10, 40, 80, 80, 40, 80, 40, 80, 240, 240, 240, 480, 240, 80, 240, 240, 80, 80, 320, 320, 480, 960, 480, 320, 960, 960, 320, 80, 320, 480, 320, 80, 32, 160, 160, 320, 640, 320, 320, 960, 960, 320, 160, 640, 960, 640, 160, 32, 160, 320, 320, 160, 32

Offset

0,3

Comments

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

The sum of the numbers in each cell of the pentatope is 7^n (A000420).

Links

Table of n, a(n) for n=0..125.

Formula

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

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

Example

The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 2401. It is organized as follows:
.
.               1
.
.               8
.            8     8
.
.              24
.           48    48
.        24    48    24
.
.              32
.           96    96
.        96   192    96
.     32    96    96    32
.
.              16
.           64    64
.        96   192    96
.     64   192   192    64
.  16    64    96    64    16

Maple

p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +2*p(i-1, j, k, l-1) +2*p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356

Prog

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

Crossrefs

Cf. A000420, A189225, A261358, A261359.

Sequence in context: A071458 A247719 A131308 * A372014 A350587 A184242

Adjacent sequences: A261357 A261358 A261359 * A261361 A261362 A261363

Keyword

nonn,tabf,walk,less

Author

Dimitri Boscainos, Aug 16 2015

Status

approved