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

1, -1, 3, -1, 15, -3, 63, -9, 135, -15, 99, -9, 12285, -945, 405, -27, 6885, -405, 161595, -8505, 1403325, -66825, 419175, -18225, 24877125, -995085, 229635, -8505, 528525, -18225, 26101845, -841995, 214708725, -6506325, 1148175, -32805, 31479513975, -850797675

Offset: 0

Author

Gregory Gerard Wojnar, Jul 17 2017

Keywords

• sign

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.

Crossrefs

Cf. A053657, A100655.

Programs

Mathematica

a[n_] := NorlundB[n, x] // Together // Numerator // Coefficient[#, x, n]&;
Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Jun 30 2019 *)

Sage

[A100655_row(n)[n] for n in (0..37)] # Peter Luschny, Jul 01 2019