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%I #43 Oct 31 2015 15:32:11
%S 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,
%T 48,24,8,24,24,8,1,8,8,8,24,48,48,24,48,24,32,96,96,96,192,96,32,96,
%U 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
%N Pentatope of coefficients in expansion of (1 + 2*x + 2*y + 2*z)^n.
%C 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).
%C The sum of the numbers in each cell of the pentatope is 7^n (A000420).
%F 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.
%F T(n,i,j,k) = 2^i*binomial(n,i)*binomial(i,j)*binomial(j,k). - _Dimitri Boscainos_, Aug 21 2015
%e 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:
%e .
%e . 1
%e .
%e . 8
%e . 8 8
%e .
%e . 24
%e . 48 48
%e . 24 48 24
%e .
%e . 32
%e . 96 96
%e . 96 192 96
%e . 32 96 96 32
%e .
%e . 16
%e . 64 64
%e . 96 192 96
%e . 64 192 192 64
%e . 16 64 96 64 16
%p p:= proc(i, j, k, l) option remember;
%p if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
%p elif {i, j, k, l}={0} then 1
%p 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)
%p fi
%p end:
%p seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
%p # Adapted from _Alois P. Heinz_'s Maple program for A261356
%o (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
%Y Cf. A000420, A189225, A261358, A261359.
%K nonn,tabf,walk,less
%O 0,3
%A _Dimitri Boscainos_, Aug 16 2015
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