OFFSET
0,3
COMMENTS
Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See arXiv:1706.08381 [math,GM], 2017.] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi(D)*g_D(N) where chi(D) := (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). The coefficients of the g_D(N) are polynomials in D of the form k_n(D)=(1/Q(n))*(D+t(n))^delta(n)*D^chi(n+1)*u_n(D) where Q(n)=A053657(n), t(n):=2 ceiling(n/2)+1, delta(n):= (1 if n is odd, 2 if n is even). The leading coefficients of u_n(D) are a(n).
LINKS
Gregory Gerard Wojnar, Table of n, a(n) for n = 0..187
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 459.
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.
MATHEMATICA
a[n_] := NorlundB[n, x] // Together // Numerator // Coefficient[#, x, n]&;
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 30 2019 *)
PROG
(Sage)
[A100655_row(n)[n] for n in (0..37)] # Peter Luschny, Jul 01 2019
CROSSREFS
KEYWORD
sign
AUTHOR
Gregory Gerard Wojnar, Jul 17 2017
STATUS
approved