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).
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
KEYWORD
nonn,tabf,walk,less
AUTHOR
Dimitri Boscainos, Aug 16 2015
STATUS
approved