Real-rooted atlas

A290316 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-29 - 30\,x + 18\,n + 18\,n x\right)\,P_{n-1}(x) + \left(-6 - 15\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27613
313922890457
44609111428694216026605
5214297684744798876399708712802168905
612820999509522707919751659811912217394282993760227515885
79379232834458531852866246664687658794004841567964137022840015268176029189279305
881109986433448826839710104666968017013290107248809878108938204269548938329836280360091609056004427906728045

All 15 rows

Technical details
Node id
n-bdb6efcdc16252c4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0d53f5dfcd557f8e

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:               7,                6;
   3:             139,              228,                90;
   4:            4609,            11142,              8694,              2160;
   5:          214297,           684744,            798876,            399708,             71280;
   6:        12820999,         50952270,          79197516,          59811912,          21739428,           2993760;
   7:       937923283,       4458531852,        8662466646,        8765879400,        4841567964,        1370228400,         152681760;
   8:     81109986433,     448826839710,     1046669680170,     1329010724880,      987810893820,      426954893832,       98362803600,       9160905600;
   9:   8094667157329,   51108049132896,   139090485294000,   212565825877140,   198847674546300,   116052168719376,    40989918282984,    7928804037600,    632102486400;
  10: 915646445959303, 6495808932145206, 20209998782523816, 36119577412673640, 40759169905020240, 30012962662763352, 14352852961217304, 4269416221548864, 709401128349600, 49303993939200;
%F Formula
The row polynomials satisfy P(n,x) = (18*n-29+(18*n-30)*x)*P(n-1,x) - (6+15*x+9*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[(18*n - 29 + (18*n - 30)*x)*P[n - 1, x] - (6 + 15*x + 9*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] = (18*n - 29 + (18*n - 30)*x)*v[i - 1] - (6 + 15*x + 9*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

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