Real-rooted atlas

A173018 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 6\,x - 3\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-x - 2\,x^{2} - x^{3} + n x + 2\,n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(-3 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3325
41214329
560100454209
636078054012451809
72520672063002408315618269
82016063840756004082496607627210853

All 15 rows

Technical details
Node id
n-91d01f3e1f751394 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ef5562416e43169f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       1;
   3:       3,       2;
   4:      12,      14,        3;
   5:      60,     100,       45,        4;
   6:     360,     780,      540,      124,       5;
   7:    2520,    6720,     6300,     2408,     315,       6;
   8:   20160,   63840,    75600,    40824,    9660,     762,      7;
   9:  181440,  665280,   952560,   667296,  233100,   36300,   1785,    8;
  10: 1814400, 7560000, 12700800, 10956960, 5103000, 1227780, 130620, 4088, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-4*n+3+(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-2,x) + ((n-1)*x+(2*n-2)*x^2+(n-1)*x^3)*d/dx P(n-2,x) + (2*n-3+(2*n-4)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 + (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 2, x] + ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*D[P[n - 2, x], x] + (2*n - 3 + (2*n - 4)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^2 - 4*n + 3 + (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 2] + ((n - 1)*x + (2*n - 2)*x^2 + (n - 1)*x^3)*deriv(v[i - 2], x) + (2*n - 3 + (2*n - 4)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.