A274496 → Veronese 2 (j=1)

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0					0
1					0
2	2				2
3	4				4
4	8				8
5	16				16
6	24	8			32
7	48	16			64
8	64	64			128
9	128	128			256
10	160	320	32		512
11	320	640	64		1024
12	384	1280	384		2048
13	768	2560	768		4096
14	896	4480	2688	128	8192

Generated b-files: A155559 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-68dd2d7821e2eea2 · revision 1 · JSON
Definition source: derived:veronese_section_2_1
Fingerprint: f5-0d8cee4552941fda

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3 (irregular)
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 ?

A050732: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A090129: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A155559: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A034008: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A118655: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A131577: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A011782: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A000079: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:   0;
  1:   0;
  2:   2;
  3:   4;
  4:   8;
  5:  16;
  6:  24,   8;
  7:  48,  16;
  8:  64,  64;
  9: 128, 128;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 4;
P[n_, x_] := P[n, x] = Expand[4*P[n - 2, x] - (4 - 4*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] = 0; v[2] = 0; v[3] = 2; v[4] = 4; for(n = 4, N, my(i = n + 1); v[i] = 4*v[i - 2] - (4 - 4*x)*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-52785358c149205f
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-52785358c149205f
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-52785358c149205f

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-52785358c149205f

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-ec4171dcd40223e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A152198 → Veronese 2 (j=1)

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