Real-rooted atlas

A166343 → γ

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(1 + 2\,n x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 10\,x\)

Rows ?

nx0x1x2x3x4Pn(1)
111011
212425
315448103
41116360477
5124217767202739
614967264998417745
711006267528352019968131247
812028923285480964339201076373

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-17caa2f11eb7e9b8 · revision 1 · JSON
Definition source
derived:gamma_polynomial
Fingerprint
f6-fddd19e8fd49984d

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 1)
  • 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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,   10;
   2: 1,   24;
   3: 1,   54,     48;
   4: 1,  116,    360;
   5: 1,  242,   1776,      720;
   6: 1,  496,   7264,     9984;
   7: 1, 1006,  26752,    83520,    19968;
   8: 1, 2028,  92328,   548096,   433920;
   9: 1, 4074, 305376,  3115664,  5458176,   867840;
  10: 1, 8168, 981312, 16127168, 52216192, 27039744;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 10*x.
%t Mathematica
P[1, x_] := 1 + 10*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 10*x; for(n = 2, N, my(i = n); v[i] = (1 + 2*n*x)*v[i - 1] + (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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-d938b07605a9ed19
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-d938b07605a9ed19
  • Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-d938b07605a9ed19

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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