Real-rooted atlas

A099093 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-7 - 11\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
212315
3451863
4171999279
5648486811215
624572268567275319
7931510206340232423247

All 15 rows

Technical details
Node id
n-4e5a71131e71d635 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-beb5f599920f9d95

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:      3;
  2:     12,      3;
  3:     45,     18;
  4:    171,     99,     9;
  5:    648,    486,    81;
  6:   2457,   2268,   567,    27;
  7:   9315,  10206,  3402,   324;
  8:  35316,  44793, 18711,  2754,   81;
  9: 133893, 192942, 96957, 19440, 1215;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n-2+(2*n-2)*x)*P(n-1,x) - (7+11*x+4*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[Together[((5*n - 2 + (2*n - 2)*x)*P[n - 1, x] - (7 + 11*x + 4*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((5*n - 2 + (2*n - 2)*x)*v[i - 1] - (7 + 11*x + 4*x^2)*deriv(v[i - 1], x))/(n)); 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
  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