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A261358 Pentatope of coefficients in expansion of (1 + x + y + 2*z)^n.

%I A261358 #42 Aug 29 2015 00:17:47
%S A261358 1,1,1,1,2,1,2,4,1,4,4,1,3,3,6,3,6,12,3,12,12,1,3,6,3,12,12,1,6,12,8,
%T A261358 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,
%U A261358 24,48,32,1,8,24,32,16,1,5,5,10,10,20,40,10,40,40,10,30,60,30,120,120,30,120,240,160,5,40,120,160,80,1,5,10,10,40,40,10,60,120,80,5,40,120,160,80,1,10,40,80,80,32
%N A261358 Pentatope of coefficients in expansion of (1 + x + y + 2*z)^n.
%C A261358 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).
%C A261358 The sum of the numbers in each cell of the pentatope is 5^n (A000351).
%C A261358 The sum of the antidiagonals of each triangle in each slice gives A261357. - _Dimitri Boscainos_, Aug 21 2015
%F A261358 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.
%F A261358 T(n,i,j,k) = 2^k*binomial(n,i)*binomial(i,j)*binomial(j,k). - _Dimitri Boscainos_, Aug 21 2015
%e A261358 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:
%e A261358 .           1
%e A261358 .
%e A261358 .           4
%e A261358 .         4   8
%e A261358 .
%e A261358 .           6
%e A261358 .         12  24
%e A261358 .        6  24  24
%e A261358 .
%e A261358 .           4
%e A261358 .        12  24
%e A261358 .      12  48  48
%e A261358 .     4  24  48  32
%e A261358 .
%e A261358 .          1
%e A261358 .        4   8
%e A261358 .      6   24  24
%e A261358 .    4  24   48   32
%e A261358 .  1   8   24   32   16
%p A261358 p:= proc(i, j, k, l) option remember;
%p A261358       if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
%p A261358     elif {i, j, k, l}={0} then 1
%p A261358     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)
%p A261358       fi
%p A261358     end:
%p A261358 seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
%p A261358 # Adapted from _Alois P. Heinz_'s Maple program for A261356
%Y A261358 Cf. A000351, A189225, A261357, A261359, A261360.
%K A261358 nonn,tabf,walk,less
%O A261358 0,5
%A A261358 _Dimitri Boscainos_, Aug 16 2015