A245171 -id:A245171 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 13 2014

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)

			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;
...

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.

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

Row sums are given by A005790.

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 *)

A245167 Triangle read by rows: coefficients of the polynomials A_{1,4}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 7, 19, 7, 0, 1, 5, 5, 1, 0, 91, 707, 1311, 707, 91, 0, 52, 570, 1655, 1655, 570, 52

Offset: 0

N. J. A. Sloane, Jul 12 2014

			Triangle begins:
1,
0,1,
0,1,1,
0,7,19,7,
0,1,5,5,1,
0,91,707,1311,707,91,
0,52,570,1655,1655,570,52,
...

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.

Cf. A245168, A245169, A245170, A245171, A245172, A245173.

A245168 Triangle read by rows: coefficients of the polynomials A_{1,5}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 8, 3, 0, 13, 63, 63, 13, 0, 663, 4938, 9038, 4938, 663, 0, 17, 177, 502, 502, 177, 17

Offset: 0

N. J. A. Sloane, Jul 13 2014

			Triangle begins:
1,
0,1,
0,1,1,
0,3,8,3,
0,13,63,63,13,
0,663,4938,9038,4938,663,
0,17,177,502,502,177,17,
...

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.

Cf. A245167, A245169, A245170, A245171, A245172, A245173.

A245169 Triangle read by rows: coefficients of the polynomials A_{1,6}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 11, 29, 11, 0, 4, 19, 19, 4, 0, 44, 319, 579, 319, 44, 0, 13, 131, 366, 366, 131, 13

Offset: 0

N. J. A. Sloane, Jul 13 2014

			Triangle begins:
1,
0,1,
0,1,1,
0,11,29,11,
0,4,19,19,4,
0,44,319,579,319,44,
0,13,131,366,366,131,13,
...

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.

Cf. A245167, A245168, A245170, A245171, A245172, A245173.

A245170 Triangle read by rows: coefficients of the polynomials A_{1,7}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 13, 34, 13, 0, 19, 89, 89, 19, 0, 1235, 8788, 15858, 8788, 1235, 0, 155, 1527, 4222, 4222, 1527, 155

Offset: 0

N. J. A. Sloane, Jul 13 2014

			Triangle begins:
1,
0,1,
0,1,1,
0,13,34,13,
0,19,89,89,19,
0,1235,8788,15858,8788,1235,
0,155,1527,4222,4222,1527,155,
...

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.

Cf. A245167, A245168, A245169, A245171, A245172, A245173.

A245172 Triangle read by rows: coefficients of the polynomials A_{2,5}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 2, 5, 2, 0, 28, 200, 363, 200, 28, 0, 77, 1056, 4263, 6650, 4263, 1056, 77, 0, 1001, 22088, 154739, 464126, 662927, 464126, 154739, 22088, 1001, 0, 1768, 56729, 605607, 2933823, 7292331, 9825279, 7292331, 2933823, 605607, 56729, 1768

Offset: 0

N. J. A. Sloane, Jul 13 2014

			Triangle begins:
1,
0,1,
0,2,5,2,
0,28,200,363,200,28,
0,77,1056,4263,6650,4263,1056,77,
0,1001,22088,154739,464126,662927,464126,154739,22088,1001,
0,1768,56729,605607,2933823,7292331,9825279,7292331,2933823,605607,56729,1768,
...

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.

Cf. A245167, A245168, A245169, A245170, A245171, A245173.

cp's OEIS Frontend

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

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A245167 Triangle read by rows: coefficients of the polynomials A_{1,4}(n,k).

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A245168 Triangle read by rows: coefficients of the polynomials A_{1,5}(n,k).

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A245169 Triangle read by rows: coefficients of the polynomials A_{1,6}(n,k).

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A245170 Triangle read by rows: coefficients of the polynomials A_{1,7}(n,k).

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A245172 Triangle read by rows: coefficients of the polynomials A_{2,5}(n,k).

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