A123519 → 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

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

\(P_{0}(x) = 6\)

\(P_{1}(x) = 12 + 8\,x\)

\(P_{2}(x) = 20 + 56\,x\)

\(P_{3}(x) = 30 + 224\,x + 32\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	6								6
1	12	8							20
2	20	56							76
3	30	224	32						286
4	42	672	352						1066
5	56	1680	2112	128					3976
6	72	3696	9152	1920					14840
7	90	7392	32032	15360	512				55386
8	110	13728	96096	87040	9728				206702
9	132	24024	256256	391680	97280	2048			771420
10	156	40040	622336	1488384	680960	47104			2878980
11	182	64064	1400256	4961280	3745280	565248	8192		10744502
12	210	99008	2956096	14883840	17228288	4710400	221184		40099026
13	240	148512	5912192	40930560	68913152	30617600	3096576	32768	149651600
14	272	217056	11286912	104600320	246118400	165335040	29933568	1015808	558507376

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Technical details

Node id: n-91e348dc87e3e358 · revision 1 · JSON
Definition source: derived:veronese_section_2_1
Fingerprint: f6-aa36ca2d330cf851

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 ?

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:   6;
  1:  12,     8;
  2:  20,    56;
  3:  30,   224,     32;
  4:  42,   672,    352;
  5:  56,  1680,   2112,    128;
  6:  72,  3696,   9152,   1920;
  7:  90,  7392,  32032,  15360,   512;
  8: 110, 13728,  96096,  87040,  9728;
  9: 132, 24024, 256256, 391680, 97280, 2048;

%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) = 6, P(1,x) = 12 + 8*x, P(2,x) = 20 + 56*x, P(3,x) = 30 + 224*x + 32*x^2.

%t Mathematica

P[0, x_] := 6;
P[1, x_] := 12 + 8*x;
P[2, x_] := 20 + 56*x;
P[3, x_] := 30 + 224*x + 32*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] = 6; v[2] = 12 + 8*x; v[3] = 20 + 56*x; v[4] = 30 + 224*x + 32*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-012c8084e12ceaa2
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-012c8084e12ceaa2
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-012c8084e12ceaa2
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-012c8084e12ceaa2

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-012c8084e12ceaa2

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