A245173 Triangle read by rows: coefficients of the polynomials A_{3,4}(n,k).

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

Offset: 0

Author

N. J. A. Sloane, Jul 13 2014

Keywords

• nonn
• tabf

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)

Examples

			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;
...
		

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.

Crossrefs

Cf. A245167, A245168, A245169, A245170, A245171, A245172.

Row sums are given by A005790.

Programs

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 *)