Real-rooted atlas

A290316 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 - 5\,x + 9\,n + 3\,n x\right)\,P_{n-1}(x) + \left(9\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2617
390481139
42160221423414609
571280114588274329961214297
62993760677062827918002607484062112820999
7152681760454137840280652364482563442178630163441937923283
89160905600342364644002915708983278467826606917119201696653065490181109986433

All 15 rows

Technical details
Node id
n-ce3ecfa80ae30839 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a228985e3530ab5e

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:              6,               1;
   3:             90,              48,               1;
   4:           2160,            2214,             234,               1;
   5:          71280,          114588,           27432,             996,              1;
   6:        2993760,         6770628,         2791800,          260748,           4062,             1;
   7:      152681760,       454137840,       280652364,        48256344,        2178630,         16344,            1;
   8:     9160905600,     34236464400,     29157089832,      7846782660,      691711920,      16966530,        65490,         1;
   9:   632102486400,   2871984146400,   3187159638984,   1219804572672,   174637926180,    8851303620,    126820980,    262092,       1;
  10: 49303993939200, 265665182896800, 369150976563264, 188635513271256, 39425596747272, 3359003385600, 105253405560, 924701832, 1048518, 1;
%F Formula
The row polynomials satisfy P(n,x) = (9*n-12+(3*n-5)*x)*P(n-1,x) + (9*x-3*x^2)*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[(9*n - 12 + (3*n - 5)*x)*P[n - 1, x] + (9*x - 3*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; for(n = 2, N, my(i = n); v[i] = (9*n - 12 + (3*n - 5)*x)*v[i - 1] + (9*x - 3*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. [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