Real-rooted atlas

A063967 → Veronese 2

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-2}(x) + \left(-2 + 2\,x\right)\,P_{n-3}(x) + \left(-1 + x\right)\,P_{n-4}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 2 + x\)
\(P_{4}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3213
4358
5516122
6843960
713104461164
82123517913448

Show all 15 stored rows

Generated b-files: A028859 (row sums), A155020 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-af589024565f1365 · revision 1 · JSON
Definition source
derived:veronese_section_2_0
Fingerprint
f6-bc743ec38800df61

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 -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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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;
   2:  1;
   3:  2,    1;
   4:  3,    5;
   5:  5,   16,    1;
   6:  8,   43,    9;
   7: 13,  104,   46,   1;
   8: 21,  235,  179,  13;
   9: 34,  506,  591,  92,  1;
  10: 55, 1051, 1746, 475, 17;
%F Formula
The row polynomials satisfy P(n,x) = 2*P(n-1,x) + (1+x)*P(n-2,x) - (2-2*x)*P(n-3,x) - (1-x)*P(n-4,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x, P(4,x) = 3 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 2 + x;
P[4, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + (1 + x)*P[n - 2, x] - (2 - 2*x)*P[n - 3, x] - (1 - x)*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 3 + 5*x; for(n = 5, N, my(i = n); v[i] = 2*v[i - 1] + (1 + x)*v[i - 2] - (2 - 2*x)*v[i - 3] - (1 - x)*v[i - 4]); 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-918cdc60ffda35ef
  • 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-918cdc60ffda35ef
  • 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-918cdc60ffda35ef

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. [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
  4. [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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741