Real-rooted atlas

A133080 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 10\,x + 17\,x^{2} + 12\,x^{3} + 3\,x^{4} - n - 6\,n x - 12\,n x^{2} - 10\,n x^{3} - 3\,n x^{4}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 12\,x^{3} + 10\,x^{4} + 3\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
20011
300123
4000011
50000123
600000011
7000000123

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e57a17a57758d913 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-538c820bb67d4056

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): yes

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1, 2;
  2: 0, 0, 1;
  3: 0, 0, 1, 2;
  4: 0, 0, 0, 0, 1;
  5: 0, 0, 0, 0, 1, 2;
  6: 0, 0, 0, 0, 0, 0, 1;
  7: 0, 0, 0, 0, 0, 0, 1, 2;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 1, 2;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(6*n-10)*x+(12*n-17)*x^2+(10*n-12)*x^3+(3*n-3)*x^4)*P(n-1,x) + (x+6*x^2+12*x^3+10*x^4+3*x^5)*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[(-(n - 2 + (6*n - 10)*x + (12*n - 17)*x^2 + (10*n - 12)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] + (x + 6*x^2 + 12*x^3 + 10*x^4 + 3*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = (-(n - 2 + (6*n - 10)*x + (12*n - 17)*x^2 + (10*n - 12)*x^3 + (3*n - 3)*x^4)*v[i - 1] + (x + 6*x^2 + 12*x^3 + 10*x^4 + 3*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a36542447e34a4df
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-a36542447e34a4df
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-a36542447e34a4df

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.