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
Examples
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.
Links
- Peter Luschny, Illustration of the polynomials.
- Peter Luschny, Eulberian polynomials, A notebook companion to A342321 and A356601, 2022.
Crossrefs
Programs
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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);
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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
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).
Comments