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 A132795 Triangle of Gely numbers, read by rows. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified May 27 19:17 EDT 2023. Contains 362985 sequences. (Running on oeis4.)