A133087 → Veronese 2

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	1	1							2
3	2	4							6
4	1	6	1						8
5	2	16	6						24
6	1	15	15	1					32
7	2	36	50	8					96
8	1	28	70	28	1				128
9	2	64	196	112	10				384
10	1	45	210	210	45	1			512
11	2	100	540	672	210	12			1536
12	1	66	495	924	495	66	1		2048
13	2	144	1210	2640	1782	352	14		6144
14	1	91	1001	3003	3003	1001	91	1	8192

Computed in your browser ?

Technical details

Node id: n-75a5395154fdfd6e · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-95f614e8f81f764d

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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 ?

A122756: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 2;
  2: 1,  1;
  3: 2,  4;
  4: 1,  6,   1;
  5: 2, 16,   6;
  6: 1, 15,  15,   1;
  7: 2, 36,  50,   8;
  8: 1, 28,  70,  28,  1;
  9: 2, 64, 196, 112, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 1 + x;
P[3, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*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; v[2] = 2; v[3] = 1 + x; v[4] = 2 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] - (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 - 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-901d84379bd0989c
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-901d84379bd0989c
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-901d84379bd0989c

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-901d84379bd0989c

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): A133080 * A007318.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A007318 e-3537980ff71b1a89
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - parent_provers periodic_product e-aa3a6b002de14e0c root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image affine, shift 0 e-28a1cf3333f63fa4
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-4d02ac33f5b2fbc9 (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