A207606 → 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

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

\(P_{0}(x) = 4 + 2\,x\)

\(P_{1}(x) = 5 + 11\,x\)

\(P_{2}(x) = 6 + 36\,x + 2\,x^{2}\)

\(P_{3}(x) = 7 + 91\,x + 19\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	4	2								6
1	5	11								16
2	6	36	2							44
3	7	91	19							117
4	8	196	100	2						306
5	9	378	385	27						799
6	10	672	1210	196	2					2090
7	11	1122	3289	1015	35					5472
8	12	1782	8008	4200	324	2				14328
9	13	2717	17875	14756	2109	43				37513
10	14	4004	37180	45696	10830	484	2			98210
11	15	5733	72930	127908	46683	3795	51			257115
12	16	8008	136136	329460	175560	23276	676	2		673134
13	17	10948	243542	791350	591261	118910	6201	59		1762288
14	18	14688	419900	1790712	1817046	526240	44226	900	2	4613732

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Technical details

Node id: n-a2065984d69a8de6 · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-e40c17441104e88c

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:  4,    2;
  1:  5,   11;
  2:  6,   36,     2;
  3:  7,   91,    19;
  4:  8,  196,   100,     2;
  5:  9,  378,   385,    27;
  6: 10,  672,  1210,   196,    2;
  7: 11, 1122,  3289,  1015,   35;
  8: 12, 1782,  8008,  4200,  324,  2;
  9: 13, 2717, 17875, 14756, 2109, 43;

%F Formula

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (6-x)*P(n-2,x) + 4*P(n-3,x) - P(n-4,x) for n >= 4, with P(0,x) = 4 + 2*x, P(1,x) = 5 + 11*x, P(2,x) = 6 + 36*x + 2*x^2, P(3,x) = 7 + 91*x + 19*x^2.

%t Mathematica

P[0, x_] := 4 + 2*x;
P[1, x_] := 5 + 11*x;
P[2, x_] := 6 + 36*x + 2*x^2;
P[3, x_] := 7 + 91*x + 19*x^2;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (6 - 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] = 4 + 2*x; v[2] = 5 + 11*x; v[3] = 6 + 36*x + 2*x^2; v[4] = 7 + 91*x + 19*x^2; for(n = 4, N, my(i = n + 1); v[i] = 4*v[i - 1] - (6 - 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-958000d578c323c8
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-958000d578c323c8
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-958000d578c323c8
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-958000d578c323c8

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-958000d578c323c8

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-62ac8b7225a515c7 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3222de37fcca30b3 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