A132795 - OEIS

%I #18 Jun 04 2014 14:17:14

%S 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,

%T 0,219,1408,2367,960,85,0,1,0,466,5482,16578,14212,3418,162,1,1,0,968,

%U 19624,99330,153824,77440,11352,341,0,1,0,1981,66496,534898,1364848,1233970,389104,36829,672,1

%N Triangle of Gely numbers, read by rows.

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

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

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

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

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

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

%D Charles O. Gely, Un tableau de conversion des polynomes cyclotomiques cousin des nombres Euleriens, Preprint Univ. Paris 7, 1999.

%D Olivier Gérard, Quelques facons originales de compter les permutations, submitted to Knuth07.

%D Olivier Gérard and Karol Penson, Set partitions, Multiset permutations and bi-permutations, in preparation.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, p. 269.

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

%e Triangle starts:

%e 1;

%e 1, 0;

%e 1, 0, 1;

%e 1, 0, 5, 0;

%e 1, 0, 16, 6, 1;

%e 1, 0, 42, 56, 21, 0;

%e ...

%o (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

%Y Cf. A000296, A132796.

%K nonn,easy,tabl

%O 0,9

%A _Olivier Gérard_, Aug 31 2007