Real-rooted atlas

A144696 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
247112
383318160
416131171411360
53247312087188812520
6641611719784222682183120160
7128528138454780954978093273741181440

All 15 rows

Technical details
Node id
n-5823146f9baf5bf7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e57a087584e4898e

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:   2,     1;
  2:   4,     7,      1;
  3:   8,    33,     18,       1;
  4:  16,   131,    171,      41,       1;
  5:  32,   473,   1208,     718,      88,       1;
  6:  64,  1611,   7197,    8422,    2682,     183,       1;
  7: 128,  5281,  38454,   78095,   49780,    9327,     374,     1;
  8: 256, 16867, 190783,  621199,  689155,  264409,   30973,   757,    1;
  9: 512, 52905, 898068, 4441476, 7862124, 5296638, 1305420, 99732, 1524, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) + (x-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*x)*P[n - 1, x] + (x - 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*x)*v[i - 1] + (x - 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