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

0,9

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).

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.

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

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

Prog

(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

Crossrefs

Cf. A000296, A132796.

Sequence in context: A277529 A354133 A060338 * A277031 A085198 A339207

Adjacent sequences: A132792 A132793 A132794 * A132796 A132797 A132798

Keyword

nonn,easy,tabl

Author

Olivier Gérard, Aug 31 2007

Status

approved