Real-rooted atlas

A100861 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-8 - 8\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
37310
42518346
5817015166
633137513515856
7130317017351053844

All 15 rows

Technical details
Node id
n-98e164af94aad52d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-98f5de0f25da5386

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     1;
  2:     3,     1;
  3:     7,     3;
  4:    25,    18,     3;
  5:    81,    70,    15;
  6:   331,   375,   135,   15;
  7:  1303,  1701,   735,  105;
  8:  5937,  9268,  5250, 1260, 105;
  9: 26785, 46908, 30618, 8820, 945;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (8+8*x+2*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[(2*n - 1 + (n - 1)*x)*P[n - 1, x] - (8 + 8*x + 2*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] = (2*n - 1 + (n - 1)*x)*v[i - 1] - (8 + 8*x + 2*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

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