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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
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
Sequence in context: A277529 A354133 A060338 * A277031 A085198 A339207
KEYWORD
nonn,easy,tabl
AUTHOR
Olivier Gérard, Aug 31 2007
STATUS
approved

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)