A132795 Triangle of Gely numbers, read by rows.

1, 1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 16, 6, 1, 1, 0, 42, 56, 21, 0, 1, 0, 99, 316, 267, 36, 1, 1, 0, 219, 1408, 2367, 960, 85, 0, 1, 0, 466, 5482, 16578, 14212, 3418, 162, 1, 1, 0, 968, 19624, 99330, 153824, 77440, 11352, 341, 0, 1, 0, 1981, 66496, 534898, 1364848, 1233970, 389104, 36829, 672, 1

Offset: 0

Author

Olivier Gérard, Aug 31 2007

Keywords

• nonn
• easy
• tabl

Comments

First row is for n=0. First column is for k=0.

Sum of rows is n! = permutations of n symbols (A000142)

These numbers are related to the Eulerian numbers A(n,k).

Third Column (k=2) is A002662(n+1).

Second Diagonal (k=n-1) is A132796.

Binomial transform of this triangle gives set partitions without singletons (in a form very close to array A105794).

Examples

			Triangle starts:
1;
1, 0;
1, 0, 1;
1, 0, 5, 0;
1, 0, 16, 6, 1;
1, 0, 42, 56, 21, 0;
...
		

References

• Charles O. Gely, Un tableau de conversion des polynomes cyclotomiques cousin des nombres Euleriens, Preprint Univ. Paris 7, 1999.
• Olivier Gérard, Quelques facons originales de compter les permutations, submitted to Knuth07.
• Olivier Gérard and Karol Penson, Set partitions, Multiset permutations and bi-permutations, in preparation.
• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, p. 269.

Crossrefs

Cf. A000296, A132796.

Programs

PARI

T(n,k)= sum(j=0, k, (-1)^j*binomial(n+1, j)*sum(m=0, n, (k-j)^m)); \\ Michel Marcus, Jun 04 2014

Formula

T(n,k) = sum(j=0..k, (-1)^j*C(n+1,j)*sum(m=0..n, (k-j)^m) ).