A086645 → Veronese 2 (j=1)

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4\right)\,P_{n-1}(x) + \left(-6 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 12\,x\right)\,P_{n-3}(x) + \left(-1 + 2\,x - x^{2}\right)\,P_{n-4}(x)\)

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

\(P_{2}(x) = 1\)

\(P_{3}(x) = 6\)

\(P_{4}(x) = 15 + x\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1					0
2	1				1
3	6				6
4	15	1			16
5	28	28			56
6	45	210	1		256
7	66	924	66		1056
8	91	3003	1001	1	4096

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0698904a58ac3702 · revision 1 · JSON
Definition source: derived:veronese_section_2_1
Fingerprint: f6-f1a8c691c0697bcb

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   1;
   3:   6;
   4:  15,     1;
   5:  28,    28;
   6:  45,   210,     1;
   7:  66,   924,    66;
   8:  91,  3003,  1001,    1;
   9: 120,  8008,  8008,  120;
  10: 153, 18564, 43758, 3060, 1;

%F Formula

The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (6-2*x)*P(n-2,x) + (4+12*x)*P(n-3,x) - (1-2*x+x^2)*P(n-4,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 6, P(4,x) = 15 + x.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 6;
P[4, x_] := 15 + x;
P[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (6 - 2*x)*P[n - 2, x] + (4 + 12*x)*P[n - 3, x] - (1 - 2*x + x^2)*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] = 0; v[2] = 1; v[3] = 6; v[4] = 15 + x; for(n = 5, N, my(i = n); v[i] = 4*v[i - 1] - (6 - 2*x)*v[i - 2] + (4 + 12*x)*v[i - 3] - (1 - 2*x + x^2)*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-6956732375ae53ac
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-6956732375ae53ac
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-6956732375ae53ac
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-6956732375ae53ac

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=1, shift 0 e-6956732375ae53ac

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-1375f2f1a11f9809 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-a5e3cd5a8b6d6a1a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-1375f2f1a11f9809 (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
[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