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A132795 Triangle of Gely numbers, read by rows.

%I A132795 #18 Jun 04 2014 14:17:14
%S A132795 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 A132795 0,219,1408,2367,960,85,0,1,0,466,5482,16578,14212,3418,162,1,1,0,968,
%U A132795 19624,99330,153824,77440,11352,341,0,1,0,1981,66496,534898,1364848,1233970,389104,36829,672,1
%N A132795 Triangle of Gely numbers, read by rows.
%C A132795 First row is for n=0. First column is for k=0.
%C A132795 Sum of rows is n! = permutations of n symbols (A000142)
%C A132795 These numbers are related to the Eulerian numbers A(n,k).
%C A132795 Third Column (k=2) is A002662(n+1).
%C A132795 Second Diagonal (k=n-1) is A132796.
%C A132795 Binomial transform of this triangle gives set partitions without singletons (in a form very close to array A105794).
%D A132795 Charles O. Gely, Un tableau de conversion des polynomes cyclotomiques cousin des nombres Euleriens, Preprint Univ. Paris 7, 1999.
%D A132795 Olivier Gérard, Quelques facons originales de compter les permutations, submitted to Knuth07.
%D A132795 Olivier Gérard and Karol Penson, Set partitions, Multiset permutations and bi-permutations, in preparation.
%D A132795 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, p. 269.
%F A132795 T(n,k) = sum(j=0..k, (-1)^j*C(n+1,j)*sum(m=0..n, (k-j)^m) ).
%e A132795 Triangle starts:
%e A132795 1;
%e A132795 1, 0;
%e A132795 1, 0, 1;
%e A132795 1, 0, 5, 0;
%e A132795 1, 0, 16, 6, 1;
%e A132795 1, 0, 42, 56, 21, 0;
%e A132795 ...
%o A132795 (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 A132795 Cf. A000296, A132796.
%K A132795 nonn,easy,tabl
%O A132795 0,9
%A A132795 _Olivier Gérard_, Aug 31 2007