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A245173 Triangle read by rows: coefficients of the polynomials A_{3,4}(n,k). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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

Row sums are given by A005790.

Sequence in context: A154155 A021942 A317577 * A256273 A046620 A046619

Adjacent sequences:  A245170 A245171 A245172 * A245174 A245175 A245176

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jul 13 2014

STATUS

approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)