OFFSET
0,6
COMMENTS
From Per W. Alexandersson, Sep 05 2019: (Start)
Let F(n,0) = 1/(1-z), and F(n,k) = z^(n-1)*( d^n/dz^n F(n,k-1) ).
The n-th row is then given by the coefficients of the monic polynomial factor in the numerator of F(n,4).
The (k+1)-th entry in row n is given by the number of standard Young tableaux of rectangular shape (n,n,n,n), with exactly k descents. (Proved by G. Panova on MathOverflow, see Links.) (End)
LINKS
Per W. Alexandersson, Table of n, a(n) for n = 0..1365
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
Greta Panova, Iterated derivative and rectangular standard Young tableaux, version: 2019-09-05.
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 6, 6, 1;
0, 1, 22, 113, 190, 113, 22, 1;
0, 1, 53, 710, 3548, 7700, 7700, 3548, 710, 53, 1;
0, 1, 105, 2856, 30422, 151389, 385029, 523200, 385029, 151389, 30422, 2856, 105, 1;
...
MATHEMATICA
GG[a_, b_] := z (Product[(k)!/(a + k)!, {k, 0, b - 1}]) z^(1 - a) (1 - z)^(a b + 1) Nest[Simplify[z^(a - 1) D[#, {z, a}]] &, 1/(1 - z), b];
Table[CoefficientList[GG[a, 4] // Together, z], {a, 1, 8}] (* Per W. Alexandersson, Sep 05 2019 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jul 13 2014
STATUS
approved