Real-rooted atlas

A143495 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-19 - 8\,x - x^{2}\right)\,P_{n-2}(x) + \left(-4 - 5\,x - x^{2}\right)\,P_{n-2}'(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
39211
46430397
5427314664811
62821287094512056761
71889024752112802240195657363
81296832088521228083374045852947499969

All 15 rows

Technical details
Node id
n-64a3befbc2c27f93 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-81d0da6b9fce0040

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:       0;
   2:       1;
   3:       9,        2;
   4:      64,       30,        3;
   5:     427,      314,       66,       4;
   6:    2821,     2870,      945,     120,       5;
   7:   18890,    24752,    11280,    2240,     195,      6;
   8:  129683,   208852,   122808,   33740,    4585,    294,     7;
   9:  918081,  1758094,  1274154,  452004,   84910,   8484,   420,   8;
  10: 6721438, 14933034, 12912231, 5654544, 1371615, 189252, 14574, 576, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(19+8*x+x^2)*P(n-2,x) - (4+5*x+x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(19 + 8*x + x^2)*P[n - 2, x] - (4 + 5*x + x^2)*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + (1 + x)*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(19 + 8*x + x^2)*v[i - 2] - (4 + 5*x + x^2)*deriv(v[i - 2], x) + (9 + 2*x)*v[i - 1] + (1 + x)*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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.