OFFSET
0,5
COMMENTS
Previous name was: Triangle generated from finite differences of A130461.
Decomposition of the permutations.
From Liam Solus, Aug 23 2018: (Start)
For k>0, T(n,k) equals the number of permutations p of [n] such that the largest index i for which p(i)>p(i+1) is k; i.e., T(n,k) is the number of permutations of [n] with maximum descent being k. See Lemma 3.4 of the paper by L. Solus below.
When T(n,k) is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h^*-polynomial is the n-th Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.
(End)
LINKS
L. Solus, Simplices for numeral systems, Transactions of the American Mathematical Society. DOI: https://doi.org/10.1090/tran/7424 (2017).
FORMULA
Each term in n-th row divides n!.
Given triangle A130461 and deleting the left border (1,1,1,...) take finite differences by columns and reorient into rows.
T(n,k) = (n-k+1+0^k)*((n+1)!/(n-k+2)!) - Olivier Gérard, Aug 04 2012
EXAMPLE
First few rows of the triangle A130461 = (1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 2, 3, 1; 1, 1, 2, 6, 4, 1;...). Deleting the left border and taking finite differences at the top of each remaining column, we get the first few rows of this triangle:
1;
1, 1;
1, 2, 3;
1, 3, 8, 12;
1, 4, 15, 40, 60;
1, 5, 24, 90, 240, 360;
1, 6, 35, 168, 630, 1680, 2520;
...
MAPLE
T := (n, k) -> (n-k+1+0^k)*((n+1)!/(n-k+2)!):
seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Sep 17 2018
MATHEMATICA
Flatten[Table[Table[(n - k + 1 + 0^k)*(n + 1)!/(n - k + 2)!, {k, 0, n}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)
PROG
(PARI) {T(n, k) = if( n<1 || k>n, 0, k==1, 1, n! * (n+1-k) / (n+2-k)!)}; /* Michael Somos, Jun 25 2017 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 28 2007
EXTENSIONS
New name using a comment by Liam Solus, Peter Luschny, Sep 17 2018
STATUS
approved