Real-rooted atlas

A008292 → γ (shift +1) → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 2\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-14 - 15\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2527
317825
41098616211
56495961361381
652856258235227214167
74451360144268803968135505

All 15 rows

Technical details
Node id
n-47057799697d7ab0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1f4fdff1bda9ed4e

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:       5,       2;
  3:      17,       8;
  4:     109,      86,      16;
  5:     649,     596,     136;
  6:    5285,    6258,    2352,     272;
  7:   44513,   60144,   26880,    3968;
  8:  448861,  712558,  407904,   97792,   7936;
  9: 4836601, 8586508, 5667144, 1645312, 176896;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-3+(2*n-2)*x)*P(n-1,x) - (14+15*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[(4*n - 3 + (2*n - 2)*x)*P[n - 1, x] - (14 + 15*x + 4*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] = (4*n - 3 + (2*n - 2)*x)*v[i - 1] - (14 + 15*x + 4*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