Real-rooted atlas

A122431 → θ+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(3\,x + n + n x\right)\,P_{n}(x) = \left(x + 4\,x^{2} + n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 6\,x + 9\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
0169420
102912528
20031215636
300041518744
4000051821852
50000062124960
6000000724271068
70000000827301176

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-8ce0c552efe4fe6e · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2f0e99ca669c600a

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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 ?

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, 6, 9,  4;
  1: 0, 2, 9, 12,  5;
  2: 0, 0, 3, 12, 15,  6;
  3: 0, 0, 0,  4, 15, 18,  7;
  4: 0, 0, 0,  0,  5, 18, 21,  8;
  5: 0, 0, 0,  0,  0,  6, 21, 24,  9;
  6: 0, 0, 0,  0,  0,  0,  7, 24, 27, 10;
  7: 0, 0, 0,  0,  0,  0,  0,  8, 27, 30, 11;
  8: 0, 0, 0,  0,  0,  0,  0,  0,  9, 30, 33, 12;
  9: 0, 0, 0,  0,  0,  0,  0,  0,  0, 10, 33, 36, 13;
%F Formula
The row polynomials satisfy (n+(n+3)*x)*P(n,x) = ((n+1)*x+(n+4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 6*x + 9*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1 + 6*x + 9*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(((n + 1)*x + (n + 4)*x^2)*P[n - 1, x])/(n + (n + 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 9*x^2 + 4*x^3; for(n = 1, N, my(i = n + 1); v[i] = (((n + 1)*x + (n + 4)*x^2)*v[i - 1])/(n + (n + 3)*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-c8839bd61c0ac6a5
  • 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-c8839bd61c0ac6a5
  • 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-c8839bd61c0ac6a5

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.