cp's OEIS Frontend

A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).

%I A290030 #39 Mar 08 2020 10:35:31
%S A290030 1,-1,3,-1,15,-3,63,-9,135,-15,99,-9,12285,-945,405,-27,6885,-405,
%T A290030 161595,-8505,1403325,-66825,419175,-18225,24877125,-995085,229635,
%U A290030 -8505,528525,-18225,26101845,-841995,214708725,-6506325,1148175,-32805,31479513975,-850797675
%N A290030 Leading coefficients of numerators of Norlund's B_{nu}^(n) polynomials (Nørlund, Tafel 5, p. 459).
%C A290030 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).
%H A290030 Gregory Gerard Wojnar, <a href="/A290030/b290030.txt">Table of n, a(n) for n = 0..187</a>
%H A290030 N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> Springer 1924, p. 459.
%H A290030 G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal Peculiar Linear Mean Relationships in All Polynomials</a>, arXiv:1706.08381 [math.GM], 2017.
%t A290030 a[n_] := NorlundB[n, x] // Together // Numerator // Coefficient[#, x, n]&;
%t A290030 Table[a[n], {n, 0, 37}] (* _Jean-François Alcover_, Jun 30 2019 *)
%o A290030 (Sage)
%o A290030 [A100655_row(n)[n] for n in (0..37)] # _Peter Luschny_, Jul 01 2019
%Y A290030 Cf. A053657, A100655.
%K A290030 sign
%O A290030 0,3
%A A290030 _Gregory Gerard Wojnar_, Jul 17 2017