A272866 → Veronese 2

palindromic γ-positive nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 + 7\,x - x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	11	1						13
3	1	30	30	1					62
4	1	58	195	58	1				313
5	1	95	685	685	95	1			1562
6	1	141	1770	3989	1770	141	1		7813
7	1	196	3801	15533	15533	3801	196	1	39062

Show all 15 stored rows

Generated b-files: A081340 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-78b19c28b9f05152 · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-bc40bb5ac844b706

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A081340: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,  11,     1;
  3: 1,  30,    30,      1;
  4: 1,  58,   195,     58,      1;
  5: 1,  95,   685,    685,     95,      1;
  6: 1, 141,  1770,   3989,   1770,    141,      1;
  7: 1, 196,  3801,  15533,  15533,   3801,    196,     1;
  8: 1, 260,  7210,  46928,  86515,  46928,   7210,   260,   1;
  9: 1, 333, 12510, 119154, 356283, 356283, 119154, 12510, 333, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-7*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 7*x + x^2)*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] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 7*x + x^2)*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-864f7f13d69989fc
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-864f7f13d69989fc
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-864f7f13d69989fc

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

veronese_section r=2, j=0, shift 0 e-864f7f13d69989fc

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[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
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[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