A321591 Partitioned 2nd-order Eulerian numbers forming an "Eulerian pyramid" (tetrahedron).

1, 1, 1, 1, 1, 4, 4, 1, 4, 1, 1, 11, 11, 11, 36, 11, 1, 11, 11, 1, 1, 26, 26, 66, 196, 66, 26, 196, 196, 26, 1, 26, 66, 26, 1, 1, 57, 57, 302, 848, 302, 302, 1898, 1898, 302, 57, 848, 1898, 848, 57, 1, 57, 302, 302, 57, 1, 1, 120, 120, 1191, 3228, 1191, 2416, 13644

Offset: 0

Author

Gregory Gerard Wojnar, Nov 13 2018

Keywords

• nonn
• tabf

Comments

For N+1 = i+j+k, let P(N+1;i,j,k) = (N+1-i)*P(N;i-1,j,k) + (N+1-j)*P(N;i,j-1,k) + (N+1-k)*P(N;i,j,k-1), with P(N;i,j,k) invariant upon permutation of the indices i,j,k, also P(N;N,0,0)=1 and P(N;i,j,k) = 0 if i or j or k is negative. The indexing of these values is shown explicitly in the examples.

The row sums are the second-order Eulerian numbers, A008517; precisely, Sum_{(j,k)|j+k=N-i} P(N;i,j,k) = <> = T(N+1,i+1) of A008517. The row sum of row i=N of slice N is (N+1)!. The sum of all entries in slice N is (2*N+1)!!. The edges of the N-th triangular slice of the pyramid are row (N+1) of the first-order Eulerian triangle, A008292.

Examples

			The first few slices of the tetrahedron (and row sums) are:
  1                  (1); i=0, N=0, (j,k)=(0,0)
------------------------
   1                 (1); i=0, N=1, (j,k)=(0,0)
  1 1                (2); i=1, N=1, (j,k)=(1,0) (0,1)
------------------------
    1                (1); i=0, N=2, (j,k)=(0,0)
   4 4               (8); i=1, N=2, (j,k)=(1,0) (0,1)
  1 4 1              (6); i=2, N=2, (j,k)=(2,0) (1,1) (0,2)
------------------------
      1              (1); i=0, N=3, (j,k)=(0,0)
    11 11           (22); i=1, N=3, (j,k)=(1,0) (0,1)
   11 36 11         (58); i=2, N=3, (j,k)=(2,0) (1,1) (0,2)
  1 11 11  1        (24); i=3, N=3, (j,k)=(3,0) (2,1) (1,2) (0,3)
------------------------
         1           (1); i=0, N=4, (j,k)=(0,0)
       26 26        (52); i=1, N=4, (j,k)=(1,0) (0,1)
     66 196 66     (328); i=2, N=4, (j,k)=(2,0) (1,1) (0,2)
   26 196 196 26   (444); i=3, N=4, (j,k)=(3,0) (2,1) (1,2) (0,3)
  1  26  66  26 1  (120); i=4, N=4, (j,k)=(4,0) (3,1) (2,2) (1,3) (0,4)