OFFSET
0,5
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),(1,1,1,0) and two kinds of steps (1,1,1,1).
The sum of the numbers in each cell of the pentatope is 5^n (A000351).
The sum of the antidiagonals of each triangle in each slice gives A261357. - Dimitri Boscainos, Aug 21 2015
FORMULA
T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + T(i,j-1,k-1,l) + T(i,j-1,k,l) + T(i,j,k,l); a(i,j,k,-1)=0,...; a(0,0,0,0)=1.
T(n,i,j,k) = 2^k*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 625. It is organized as follows:
. 1
.
. 4
. 4 8
.
. 6
. 12 24
. 6 24 24
.
. 4
. 12 24
. 12 48 48
. 4 24 48 32
.
. 1
. 4 8
. 6 24 24
. 4 24 48 32
. 1 8 24 32 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) +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
CROSSREFS
KEYWORD
nonn,tabf,walk,less
AUTHOR
Dimitri Boscainos, Aug 16 2015
STATUS
approved