Real-rooted atlas

A257619 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-5 - 7\,x + 9\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
252524108
31144171658883456
4354647092841964650016154872
514185603546400298012092378071564328940456
6695094402085283202271469201067466401940593278733264632124648
740315475201411041632018886707840119407288003522583064398353956866090812852898998536

All 15 rows

Technical details
Node id
n-9d3de644c7283a2d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cc3d28c31b2705bc

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:              4,              2;
  2:             52,             52,               4;
  3:           1144,           1716,             588,               8;
  4:          35464,          70928,           41964,            6500,             16;
  5:        1418560,        3546400,         2980120,          923780,          71564,             32;
  6:       69509440,      208528320,       227146920,       106746640,       19405932,         787332,            64;
  7:     4031547520,    14110416320,     18886707840,     11940728800,     3522583064,      398353956,       8660908,          128;
  8:   270113683840,  1080454735360,   1715631998080,   1365304420480,   562947769384,   110918695888,    8079670924,     95270500,        256;
  9: 20528639971840, 92378879873280, 169736213687040, 162975308496000, 86687218537920, 24830494809120, 3394394941080, 162831934980, 1047976524, 512;
%F Formula
The row polynomials satisfy P(n,x) = (9*n-5+(9*n-7)*x)*P(n-1,x) - (9*x+9*x^2)*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[(9*n - 5 + (9*n - 7)*x)*P[n - 1, x] - (9*x + 9*x^2)*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] = (9*n - 5 + (9*n - 7)*x)*v[i - 1] - (9*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 - 1, v[i])))};
T(9)

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