A260326 Common denominator of coefficients in Nörlund's polynomial D_{2n}(x).

1, 3, 15, 63, 135, 99, 12285, 405, 6885, 161595, 1403325, 419175, 24877125, 229635, 528525, 26101845, 214708725, 1148175, 31479513975, 134336475, 23302211625, 513217002375, 374333754375, 50996192625, 25178013826875, 678264862275, 813304020375, 6122798191125

Offset: 0

Author

N. J. A. Sloane, Jul 25 2015

Keywords

• nonn
• frac

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 [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

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..100
• N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 460.
• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 460 [Annotated scanned copy of pages 144-151 and 456-463]
• G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

Crossrefs

For numerators see A260327.

Cf. A053657.

Programs

Maple

# NorlundD polynomials are defined in A260327.
seq(denom(NorlundD(2*n)(x)), n=0..27); # Peter Luschny, Jul 01 2019

Mathematica

NorlundD[nu_, n_] := (-2)^nu NorlundB[nu, n, n/2] // Simplify;
a[n_] := Module[{nb}, nb = NorlundB[2n, x]; nb/Coefficient[nb, x, 2n] // Together // Denominator];
(* or: *)
a[n_] := (2n)! SeriesCoefficient[(z/Sin[z])^x, {z, 0, 2n}] // Normal // Together // Denominator;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 01 2019 *)

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

Formula

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

Extensions

Terms a(7) and beyond from Gregory Gerard Wojnar, Jul 19 2017

a(24)-a(27) corrected by Jean-François Alcover, Jul 01 2019