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

%I A245173 #28 Sep 23 2019 14:46:40
%S A245173 1,0,1,0,1,6,6,1,0,1,22,113,190,113,22,1,0,1,53,710,3548,7700,7700,
%T A245173 3548,710,53,1,0,1,105,2856,30422,151389,385029,523200,385029,151389,
%U A245173 30422,2856,105,1
%N A245173 Triangle read by rows: coefficients of the polynomials A_{3,4}(n,k).
%C A245173 From _Per W. Alexandersson_, Sep 05 2019: (Start)
%C A245173 Let F(n,0) = 1/(1-z), and F(n,k) = z^(n-1)*( d^n/dz^n F(n,k-1) ).
%C A245173 The n-th row is then given by the coefficients of the monic polynomial factor in the numerator of F(n,4).
%C A245173 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)
%H A245173 Per W. Alexandersson, <a href="/A245173/b245173.txt">Table of n, a(n) for n = 0..1365</a>
%H A245173 J. Agapito, <a href="https://dx.doi.org/10.1016/j.laa.2014.03.018">On symmetric polynomials with only real zeros and nonnegative gamma-vectors</a>, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
%H A245173 Greta Panova, <a href="https://mathoverflow.net/q/339893">Iterated derivative and rectangular standard Young tableaux</a>, version: 2019-09-05.
%e A245173 Triangle begins:
%e A245173   1;
%e A245173   0, 1;
%e A245173   0, 1, 6, 6, 1;
%e A245173   0, 1, 22, 113, 190, 113, 22, 1;
%e A245173   0, 1, 53, 710, 3548, 7700, 7700, 3548, 710, 53, 1;
%e A245173   0, 1, 105, 2856, 30422, 151389, 385029, 523200, 385029, 151389, 30422, 2856, 105, 1;
%e A245173 ...
%t A245173 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];
%t A245173 Table[CoefficientList[GG[a, 4] // Together, z], {a, 1, 8}] (* _Per W. Alexandersson_, Sep 05 2019 *)
%Y A245173 Cf. A245167, A245168, A245169, A245170, A245171, A245172.
%Y A245173 Row sums are given by A005790.
%K A245173 nonn,tabf
%O A245173 0,6
%A A245173 _N. J. A. Sloane_, Jul 13 2014