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A342321 T(n, k) = A343277(n)*[x^k] p(n, x) where p(n, x) = (1/(n+1))*Sum_{k=0..n} (-1)^k*E1(n, k)*x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.

%I A342321 #17 Aug 10 2024 05:10:39
%S A342321 1,0,1,0,-1,2,0,1,-4,3,0,-3,22,-33,12,0,1,-13,33,-26,5,0,-5,114,-453,
%T A342321 604,-285,30,0,5,-200,1191,-2416,1985,-600,35,0,-35,2470,-21465,62476,
%U A342321 -78095,42930,-8645,280,0,14,-1757,21912,-88234,156190,-132351,51128,-7028,126
%N A342321 T(n, k) = A343277(n)*[x^k] p(n, x) where p(n, x) = (1/(n+1))*Sum_{k=0..n} (-1)^k*E1(n, k)*x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.
%C A342321 Conjecture: For even n >= 6 p(n, x)/x and for odd n >= 3 p(n, x)/(x^2 - x) is irreducible.
%H A342321 Peter Luschny, <a href="/A342321/a342321.jpg">Illustration of the polynomials.</a>
%H A342321 Peter Luschny, <a href="https://github.com/PeterLuschny/EulerianSecondOrderNumbers/blob/main/EulberianPolynomials.ipynb">Eulberian polynomials</a>, A notebook companion to A342321 and A356601, 2022.
%F A342321 An alternative representation of the sequence of rational polynomials is:
%F A342321 p(n, x) = Sum_{k=1..n} x^k*k!*Sum_{j=0..k} (-1)^(n-j)*Stirling2(n, j)/((k - j)!(n - j + 1)*binomial(n + 1, j)) for n >= 1 and p(0, x) = 1.
%F A342321 (Sum_{k = 0..n} T(n, k)) / A343277(n) = Bernoulli(n, 1).
%e A342321 Triangle starts:
%e A342321 [n]                T(n, k)                      A343277(n)
%e A342321 ----------------------------------------------------------
%e A342321 [0] 1;                                                 [1]
%e A342321 [1] 0,  1;                                             [2]
%e A342321 [2] 0, -1,     2;                                      [6]
%e A342321 [3] 0,  1,    -4,     3;                              [12]
%e A342321 [4] 0, -3,    22,   -33,    12;                       [60]
%e A342321 [5] 0,  1,   -13,    33,   -26,     5;                [30]
%e A342321 [6] 0, -5,   114,  -453,   604,  -285,    30;        [210]
%e A342321 [7] 0,  5,  -200,  1191, -2416,  1985,  -600,  35;   [280]
%e A342321 .
%e A342321 The coefficients of the polynomials p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A343277(n) for the first few n:
%e A342321 [0] 1;
%e A342321 [1] 0,   1/2;
%e A342321 [2] 0,  -1/6,    1/3;
%e A342321 [3] 0,  1/12,   -1/3,    1/4;
%e A342321 [4] 0, -1/20,   11/30, -11/20,    1/5;
%e A342321 [5] 0,  1/30,  -13/30,  11/10,  -13/15,  1/6.
%p A342321 CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
%p A342321 E1 := (n, k) -> combinat:-eulerian1(n, k):
%p A342321 poly := n -> (1/(n+1))*add((-1)^k*E1(n,k)*x^(n-k)/binomial(n,k), k=0..n):
%p A342321 Trow := n -> denom(poly(n))*CoeffList(poly(n)): seq(Trow(n), n = 0..9);
%t A342321 Poly342321[n_, x_] := If[n == 0, 1, Sum[x^k*k!*Sum[(-1)^(n - j)*StirlingS2[n, j] /((k - j)!(n - j + 1) Binomial[n + 1, j]), {j, 0, k}], {k, 1, n}]];
%t A342321 Table[A343277[n] CoefficientList[Poly342321[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten
%Y A342321 Cf. A343277, A123125.
%Y A342321 Sequences of rational polynomials p(n, x) with p(n, 1) = Bernoulli(n, 1):
%Y A342321 A342312/A342313, A342321/A342320, A342322/A064538.
%K A342321 sign,tabl,frac
%O A342321 0,6
%A A342321 _Peter Luschny_, Mar 09 2021