A261359 Pentatope of coefficients in expansion of (1 + x + 2*y + 2*z)^n.

1, 1, 1, 2, 2, 1, 2, 4, 4, 1, 4, 4, 4, 8, 4, 1, 3, 6, 6, 3, 12, 12, 12, 24, 12, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 4, 8, 8, 6, 24, 24, 24, 48, 24, 4, 24, 24, 48, 96, 48, 32, 96, 96, 32, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 5, 10, 10, 10, 40, 40, 40, 80, 40, 10, 60, 60, 120, 240, 120, 80, 240, 240, 80, 5, 40, 40, 120, 240, 120, 160, 480, 480, 160, 80, 320, 480, 320, 80, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32

Offset

0,4

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), (1,1,0,0) and two kinds of steps (1,1,1,0) and (1,1,1,1).

The sum of the numbers in each cell of the pentatope is 6^n (A000400).

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) + 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^j*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 1296. It is organized as follows:
.
.           1
.
.           4
.         8   8
.
.           6
.        24  24
.      24  48  24
.
.           4
.        24  24
.      48  96  48
.    32  96  96  32
.
.           1
.         8   8
.      24  48  24
.    32  96  96  32
.  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) +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^j*binomial(n, i)*binomial(i, j)*binomial(j, k), ", ")); ); ); ); } \\ Michel Marcus, Oct 07 2015

Crossrefs

Cf. A000400, A189225, A261358, A261360.

Sequence in context: A229219 A048299 A232084 * A217680 A144218 A098691

Adjacent sequences: A261356 A261357 A261358 * A261360 A261361 A261362

Keyword

nonn,tabf,walk,less

Author

Dimitri Boscainos, Aug 16 2015

Status

approved