OFFSET
0,6
COMMENTS
Conjecture: For even n >= 6 p(n, x)/x and for odd n >= 3 p(n, x)/(x^2 - x) is irreducible.
LINKS
Peter Luschny, Illustration of the polynomials.
FORMULA
An alternative representation of the sequence of rational polynomials is:
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.
(Sum_{k = 0..n} T(n, k)) / A343277(n) = Bernoulli(n, 1).
EXAMPLE
Triangle starts:
[n] T(n, k) A343277(n)
----------------------------------------------------------
[0] 1; [1]
[1] 0, 1; [2]
[2] 0, -1, 2; [6]
[3] 0, 1, -4, 3; [12]
[4] 0, -3, 22, -33, 12; [60]
[5] 0, 1, -13, 33, -26, 5; [30]
[6] 0, -5, 114, -453, 604, -285, 30; [210]
[7] 0, 5, -200, 1191, -2416, 1985, -600, 35; [280]
.
The coefficients of the polynomials p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A343277(n) for the first few n:
[0] 1;
[1] 0, 1/2;
[2] 0, -1/6, 1/3;
[3] 0, 1/12, -1/3, 1/4;
[4] 0, -1/20, 11/30, -11/20, 1/5;
[5] 0, 1/30, -13/30, 11/10, -13/15, 1/6.
MAPLE
CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E1 := (n, k) -> combinat:-eulerian1(n, k):
poly := n -> (1/(n+1))*add((-1)^k*E1(n, k)*x^(n-k)/binomial(n, k), k=0..n):
Trow := n -> denom(poly(n))*CoeffList(poly(n)): seq(Trow(n), n = 0..9);
MATHEMATICA
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}]];
Table[A343277[n] CoefficientList[Poly342321[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Mar 09 2021
STATUS
approved