A342322 T(n, k) = A064538(n)*[x^k] p(n, x) where p(n, x) = 1 + Sum_{k = 0..n-1} binomial[n, k]*p(k, 1)* ((x - 1)^(n - k) - 1) / (n - k + 1) for n >= 1 and p(0, x) = 1. Triangle read by rows, for 0 <= k <= n.

1, 0, 1, 0, -1, 2, 0, 0, -1, 1, 0, 1, 1, -9, 6, 0, 0, 1, 1, -4, 2, 0, -1, -1, 6, 6, -15, 6, 0, 0, -2, -2, 5, 5, -9, 3, 0, 3, 3, -17, -17, 25, 25, -35, 10, 0, 0, 3, 3, -7, -7, 7, 7, -8, 2, 0, -5, -5, 28, 28, -38, -38, 28, 28, -27, 6, 0, 0, -10, -10, 23, 23, -21, -21, 12, 12, -10, 2

Offset: 0

Author

Peter Luschny, Mar 09 2021

Keywords

• sign
• frac
• tabl

Examples

			p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A064538(n).
[n]                T(n, k)               A064538(n)
---------------------------------------------------
[0] 1,                                         [ 1]
[1] 0,  1,                                     [ 2]
[2] 0, -1,  2,                                 [ 6]
[3] 0,  0, -1,   1,                            [ 4]
[4] 0,  1,  1,  -9,   6,                       [30]
[5] 0,  0,  1,   1,  -4,   2,                  [12]
[6] 0, -1, -1,   6,   6, -15,  6,              [42]
[7] 0,  0, -2,  -2,   5,   5, -9,   3,         [24]
[8] 0,  3,  3, -17, -17,  25, 25, -35, 10,     [90]
[9] 0,  0,  3,   3,  -7,  -7,  7,   7, -8, 2.  [20]
		

Crossrefs

Cf. A064538.

Programs

Maple

CoeffList := p -> [op(PolynomialTools:-CoefficientList(factor(p), x))]:
p := n -> add(binomial(n+1,k+1)*bernoulli(n-k, 1)*(x-1)^k, k=0..n)/(n+1):
seq(print(denom(p(n))*CoeffList(p(n))), n=0..9);

Mathematica

(* Uses the function A064538. *)
p[n_, x_] := p[n, x] = If[n == 0, 1, 1 +
Sum[Binomial[n, k] p[k, 1] ((x - 1)^(n - k) - 1) / (n - k + 1), {k, 0, n-1}]];
Table[A064538[n] CoefficientList[p[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten

Formula

(Sum_{k = 0..n} T(n, k)) / A064538(n) = Bernoulli(n, 1).