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.

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, 604, -285, 30, 0, 5, -200, 1191, -2416, 1985, -600, 35, 0, -35, 2470, -21465, 62476, -78095, 42930, -8645, 280, 0, 14, -1757, 21912, -88234, 156190, -132351, 51128, -7028, 126

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

Table of n, a(n) for n=0..54.

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

Cf. A343277, A123125.

Sequences of rational polynomials p(n, x) with p(n, 1) = Bernoulli(n, 1):

A342312/A342313, A342321/A342320, A342322/A064538.

Sequence in context: A334781 A291656 A209063 * A098689 A288515 A264583

Adjacent sequences: A342318 A342319 A342320 * A342322 A342323 A342324

Keyword

sign,tabl,frac

Author

Peter Luschny, Mar 09 2021

Status

approved