Real-rooted atlas

A225467 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 18\,x + 23\,x^{2} + 8\,x^{3} - 4\,n x - 8\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23710
395865132
42739710497432216
58126441240619508992144560
6243177791325263477343818711501511030304
77291212941364031532912494903117927150252659326759232
82187836665138951997614694919579528126072962717461997346615783768641664

All 15 rows

Technical details
Node id
n-d713cd65d4f1a512 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8d7e707202f48734

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,        7;
   3:     9,       58,         65;
   4:    27,      397,       1049,         743;
   5:    81,     2644,      12406,       19508,        9921;
   6:   243,    17779,     132526,      347734,      381871,       150151;
   7:   729,   121294,    1364031,     5329124,     9490311,      7927150,      2526593;
   8:  2187,   836665,   13895199,    76146949,   195795281,    260729627,    174619973,     46615783;
   9:  6561,  5810728,  141909308,  1052648280,  3670647206,   6932178328,   7319058812,   4073991528,    933072513;
  10: 19683, 40511071, 1462010060, 14349840044, 65283825242, 164081666386, 242866907548, 211445950108, 100398804827, 20093861895;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3-(4*n-18)*x-(8*n-23)*x^2-(4*n-8)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3 - (4*n - 18)*x - (8*n - 23)*x^2 - (4*n - 8)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3 - (4*n - 18)*x - (8*n - 23)*x^2 - (4*n - 8)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.