Real-rooted atlas

A128099 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + n\right)\,P_{n}(x) = \left(-2 - 20\,x + n + 4\,n x\right)\,P_{n-1}(x) + \left(-x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 0\)
\(P_{4}(x) = 8\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
20
30
488
52424
6484896
780192272
8120480192792
91689609602088
10224168028806405424
1128826886720384013536
1236040321344013440192033192
13440576024192358401344079672
1452879204032080640537605376188544
Technical details
Node id
n-4f19d6ed4b208f66 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b4388d7923690e8a

Observed on the stored rows ?

  • Degrees: -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5 (half affine: slope 1, intercept -4)
  • 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:   0;
  1:   0;
  2:   0;
  3:   0;
  4:   8;
  5:  24;
  6:  48,  48;
  7:  80, 192;
  8: 120, 480, 192;
  9: 168, 960, 960;
%F Formula
The row polynomials satisfy (n-4)*P(n,x) = (n-2+(4*n-20)*x)*P(n-1,x) - (x+8*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 0;
P[4, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 20)*x)*P[n - 1, x] - (x + 8*x^2)*D[P[n - 1, x], x])/(n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 0; v[5] = 8; for(n = 5, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 20)*x)*v[i - 1] - (x + 8*x^2)*deriv(v[i - 1], x))/(n - 4)); 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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.