A342322 - OEIS

%I #14 Apr 10 2021 20:07:54

%S 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,

%T -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,

%U -5,-5,28,28,-38,-38,28,28,-27,6,0,0,-10,-10,23,23,-21,-21,12,12,-10,2

%N 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.

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

%e p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A064538(n).

%e [n]                T(n, k)               A064538(n)

%e ---------------------------------------------------

%e [0] 1,                                         [ 1]

%e [1] 0,  1,                                     [ 2]

%e [2] 0, -1,  2,                                 [ 6]

%e [3] 0,  0, -1,   1,                            [ 4]

%e [4] 0,  1,  1,  -9,   6,                       [30]

%e [5] 0,  0,  1,   1,  -4,   2,                  [12]

%e [6] 0, -1, -1,   6,   6, -15,  6,              [42]

%e [7] 0,  0, -2,  -2,   5,   5, -9,   3,         [24]

%e [8] 0,  3,  3, -17, -17,  25, 25, -35, 10,     [90]

%e [9] 0,  0,  3,   3,  -7,  -7,  7,   7, -8, 2.  [20]

%p CoeffList := p -> [op(PolynomialTools:-CoefficientList(factor(p), x))]:

%p p := n -> add(binomial(n+1,k+1)*bernoulli(n-k, 1)*(x-1)^k, k=0..n)/(n+1):

%p seq(print(denom(p(n))*CoeffList(p(n))), n=0..9);

%t (* Uses the function A064538. *)

%t p[n_, x_] := p[n, x] = If[n == 0, 1, 1 +

%t Sum[Binomial[n, k] p[k, 1] ((x - 1)^(n - k) - 1) / (n - k + 1), {k, 0, n-1}]];

%t Table[A064538[n] CoefficientList[p[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten

%Y Cf. A064538.

%K sign,frac,tabl

%O 0,6

%A _Peter Luschny_, Mar 09 2021