Real-rooted atlas

A136572 → basis A013609 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(3\,n - 2\,n x - 3\,n^{2} + 2\,n^{2} x\right)\,P_{n-2}(x) + \left(4\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21832858
3162492312481014
4194484489408345638423640
529160165360273600172800441603840688920
652488036748808210880768384033062406451204608024091920
711022480918590402607696003313094402090995206757632010644480645120982926000

All 15 rows

Technical details
Node id
n-55a69da1c48f482e · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          3,           2;
  2:         18,          32,            8;
  3:        162,         492,          312,           48;
  4:       1944,        8448,         9408,         3456,          384;
  5:      29160,      165360,       273600,       172800,        44160,         3840;
  6:     524880,     3674880,      8210880,      7683840,      3306240,       645120,        46080;
  7:   11022480,    91859040,    260769600,    331309440,    209099520,     67576320,     10644480,      645120;
  8:  264539520,  2557255680,   8846530560,  14402949120,  12297277440,   5769953280,   1486356480,   196116480,   10321920;
  9: 7142567040, 78568600320, 321407170560, 643731701760, 704498135040, 444583157760, 164799774720, 35208069120, 3994583040, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = -(3*n^2-3*n-(2*n^2-2*n)*x)*P(n-2,x) + (4*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3*n^2 - 3*n - (2*n^2 - 2*n)*x)*P[n - 2, x] + (4*n + 2*n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n^2 - 3*n - (2*n^2 - 2*n)*x)*v[i - 2] + (4*n + 2*n*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?

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.