A123519 → Veronese 2

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 4\,x\)

\(P_{1}(x) = 1 + 20\,x\)

\(P_{2}(x) = 1 + 60\,x + 16\,x^{2}\)

\(P_{3}(x) = 1 + 140\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	1	4								5
1	1	20								21
2	1	60	16							77
3	1	140	144							285
4	1	280	720	64						1065
5	1	504	2640	832						3977
6	1	840	7920	5824	256					14841
7	1	1320	20592	29120	4352					55385
8	1	1980	48048	116480	39168	1024				206701
9	1	2860	102960	396032	248064	21504				771421
10	1	4004	205920	1188096	1240320	236544	4096			2878981
11	1	5460	388960	3224832	5209344	1813504	102400			10744501
12	1	7280	700128	8062080	19100928	10881024	1331200	16384		40099025
13	1	9520	1209312	18811520	62760192	54405120	11980800	475136		149651601
14	1	12240	2015520	41385344	188280576	235755520	83865600	7127040	65536	558507377

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Technical details

Node id: n-77ff1b7e9c0102bd · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-6d3aab3522b27b2a

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): no

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,    4;
  1: 1,   20;
  2: 1,   60,     16;
  3: 1,  140,    144;
  4: 1,  280,    720,     64;
  5: 1,  504,   2640,    832;
  6: 1,  840,   7920,   5824,    256;
  7: 1, 1320,  20592,  29120,   4352;
  8: 1, 1980,  48048, 116480,  39168,  1024;
  9: 1, 2860, 102960, 396032, 248064, 21504;

%F Formula

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (6-4*x)*P(n-2,x) + 4*P(n-3,x) - P(n-4,x) for n >= 4, with P(0,x) = 1 + 4*x, P(1,x) = 1 + 20*x, P(2,x) = 1 + 60*x + 16*x^2, P(3,x) = 1 + 140*x + 144*x^2.

%t Mathematica

P[0, x_] := 1 + 4*x;
P[1, x_] := 1 + 20*x;
P[2, x_] := 1 + 60*x + 16*x^2;
P[3, x_] := 1 + 140*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (6 - 4*x)*P[n - 2, x] + 4*P[n - 3, x] - P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x; v[2] = 1 + 20*x; v[3] = 1 + 60*x + 16*x^2; v[4] = 1 + 140*x + 144*x^2; for(n = 4, N, my(i = n + 1); v[i] = 4*v[i - 1] - (6 - 4*x)*v[i - 2] + 4*v[i - 3] - v[i - 4]); 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-01d67bf255e61a74
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-01d67bf255e61a74
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-01d67bf255e61a74
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-01d67bf255e61a74

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-01d67bf255e61a74

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023