A153359 - OEIS

A153359

Scaled coefficients of the M. O. Rubinstein polynomials.

1, -1, 1, -2, -1, 3, -2, -1, 2, 1, -152, -78, 125, 90, 15, -216, -114, 157, 135, 35, 3, -41424, -22444, 27552, 26551, 8505, 1197, 63, -66000, -36620, 40976, 42917, 15652, 2814, 252, 9, -13037952, -7390832, 7652084, 8557940, 3414775, 714840, 83790, 5220, 135, -21995904

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OFFSET

0,4

COMMENTS

The polynomials alpha_{k}(s) are defined in formula (1.4) in the paper cited below. The coefficients are in ascending order.

LINKS

Table of n, a(n) for n=0..45.

M. O. Rubinstein, Identities for the Riemann Zeta function., arXiv:0812.2592 [math.NT]

FORMULA

The coefficients of the polynomials alpha_{k}(s)*A053657(k) where alpha_{0}(s) = 1 and alpha_{k+1}(s) = (s-1)/(k+2)-sum(j=1..k,((j-(s-1)*(k-j+1))/(k-j+2))*alpha_{j}(s))/(k+1).

EXAMPLE

alpha_{0}(t) = 1 / 1;

alpha_{1}(t) = (-1 + t) / 2;

alpha_{2}(t) = (-2 - t + 3t^2) / 24;

alpha_{3}(t) = (-2 - t + 2t^2 + t^3) / 48;

MATHEMATICA

alpha[0, _] = 1; alpha[k_, s_] := (s - 1)/(k + 1) - Sum[((j - (s - 1)*(k - j))/(k - j + 1))*alpha[j, s]/(k), {j, 1, k - 1}] // Expand;

a53657[n_] := Product[p^Sum[Floor[(n - 1)/((p - 1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}];

row[k_] := CoefficientList[alpha[k, t]*a53657[k + 1], t];

Table[row[k], {k, 0, 7}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

CROSSREFS

Cf. A053657.