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A321591
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Partitioned 2nd-order Eulerian numbers forming an "Eulerian pyramid" (tetrahedron).
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0
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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
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OFFSET
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0,6
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COMMENTS
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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) = <<N+1,i>> = 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.
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LINKS
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EXAMPLE
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The first few slices of the tetrahedron (and row sums) are:
1 (1); i=0, N=0, (j,k)=(0,0)
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1 (1); i=0, N=1, (j,k)=(0,0)
1 1 (2); i=1, N=1, (j,k)=(1,0) (0,1)
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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)
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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)
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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)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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