Real-rooted atlas

A133087 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 - x - 10\,x^{2} + 4\,n x + 10\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 9\,x^{2} - 10\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
21449
317161236
41824321681
51114810411248324
61126016024019264729
7115963407209126401922916
811611244811201792179210242566561
91191607842464515271686400332876826244
10120180960336080641344015360115205120102459049
11123240150062401814437632556805760039680163843072236196
1212426417607920253445913610137612672011264067584245764096531441
131273362552132004910413516827878443084849280040550422732877824122882125764
141283642912160166406419219243929676876810250241025024745472372736114688163844782969
Technical details
Node id
n-050402cbf2865684 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6c06ab9d9924dc09

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  4,   4;
  3: 1,  7,  16,  12;
  4: 1,  8,  24,  32,   16;
  5: 1, 11,  48, 104,  112,   48;
  6: 1, 12,  60, 160,  240,  192,   64;
  7: 1, 15,  96, 340,  720,  912,  640,  192;
  8: 1, 16, 112, 448, 1120, 1792, 1792, 1024,  256;
  9: 1, 19, 160, 784, 2464, 5152, 7168, 6400, 3328, 768;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-1)*x+(10*n-10)*x^2)*P(n-1,x) - (2*x+9*x^2+10*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 1)*x + (10*n - 10)*x^2)*P[n - 1, x] - (2*x + 9*x^2 + 10*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n - 1)*x + (10*n - 10)*x^2)*v[i - 1] - (2*x + 9*x^2 + 10*x^3)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Candidate equalities ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.