Real-rooted atlas

A122431 → Veronese 2 (j=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
\(P_{n}(x) = \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 3 + x\)
\(P_{1}(x) = 1 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0314
1134
20314
30134
400314
500134
6000314
7000134
80000314
90000134
1000000314
1100000134
12000000314
13000000134
140000000314

Computed in your browser ?

Technical details
Node id
n-ab55e05a7226d6fc · revision 1 · JSON
Definition source
derived:veronese_section_2_1
Fingerprint
f6-714179c45d3ad4b0

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 2)
  • 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: 3, 1;
  1: 1, 3;
  2: 0, 3, 1;
  3: 0, 1, 3;
  4: 0, 0, 3, 1;
  5: 0, 0, 1, 3;
  6: 0, 0, 0, 3, 1;
  7: 0, 0, 0, 1, 3;
  8: 0, 0, 0, 0, 3, 1;
  9: 0, 0, 0, 0, 1, 3;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-2,x) for n >= 2, with P(0,x) = 3 + x, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 3 + x;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-267d8a8f48506a34
  • 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-267d8a8f48506a34
  • 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-267d8a8f48506a34

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3