Real-rooted atlas

A008292 → γ (scale 1/2) (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

\(P_{n}(x) = \left(-1 - x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-6 - 7\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2314
39413
43927470
518916234385
611071185384342710
772819144374449620665
85435179191405368256496182830
9448821740286448146117216110561765525
1040858837545069531284417409542521441105618947950
11405331298273566866213288258256084867128349504220524325
1243584795997507565187567966039833316094307280104073363495042790000550
1350457450691227338130612213235890633539520017948570402605736161487310437938061225
146259482902716450348133717974214733610485492525034735700880635426452855862428814873104553358845750
Technical details
Node id
n-55e1345523492ef8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-03bcbd90378cd7cd

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:      9,      4;
  4:     39,     27,      4;
  5:    189,    162,     34;
  6:   1107,   1185,    384,     34;
  7:   7281,   9144,   3744,    496;
  8:  54351,  79191,  40536,   8256,   496;
  9: 448821, 740286, 448146, 117216, 11056;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (6+7*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] - (6 + 7*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] - (6 + 7*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