A260326 - OEIS

%I #79 Jul 04 2019 13:38:58

%S 1,3,15,63,135,99,12285,405,6885,161595,1403325,419175,24877125,

%T 229635,528525,26101845,214708725,1148175,31479513975,134336475,

%U 23302211625,513217002375,374333754375,50996192625,25178013826875,678264862275,813304020375,6122798191125

%N Common denominator of coefficients in Nörlund's polynomial D_{2n}(x).

%C 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 [Wojnar et al., 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). For odd n, the leading coefficients of u_n(D) are a((n+1)/2). - _Gregory Gerard Wojnar_, Jul 17 2017

%H Jean-François Alcover, <a href="/A260326/b260326.txt">Table of n, a(n) for n = 0..100</a>

%H N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> Springer 1924, p. 460.

%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924; page 460 [Annotated scanned copy of pages 144-151 and 456-463]

%H 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.

%F E.g.f. Sum_{n>=0} D_{2n}(x) y^(2n)/(2n)! = (y/sinh(y))^x. - _Max Alekseyev_, Jul 04 2019

%p # NorlundD polynomials are defined in A260327.

%p seq(denom(NorlundD(2*n)(x)), n=0..27); # _Peter Luschny_, Jul 01 2019

%t NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2] // Simplify;

%t a[n_] := Module[{nb}, nb = NorlundB[2n, x]; nb/Coefficient[nb, x, 2n] // Together // Denominator];

%t (* or: *)

%t a[n_] := (2n)! SeriesCoefficient[(z/Sin[z])^x, {z, 0, 2n}] // Normal // Together // Denominator;

%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jul 01 2019 *)

%o (PARI) { A260326(n) = my(t, Y); Y=y+O(y^(2*n+2)); t = (2*n)! * Pol( polcoeff( exp( x * log(Y/sinh(Y)) + O(x^(n+1)) ), 2*n, y ) ); denominator(content(t)); } \\ _Max Alekseyev_, Jul 04 2019

%Y For numerators see A260327.

%Y Cf. A053657.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_, Jul 25 2015

%E Terms a(7) and beyond from _Gregory Gerard Wojnar_, Jul 19 2017

%E a(24)-a(27) corrected by _Jean-François Alcover_, Jul 01 2019