A135838 → Veronese 2

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 4\,x\right)\,P_{n-2}(x) + \left(-4 + 8\,x - 4\,x^{2}\right)\,P_{n-4}(x)\)

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

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

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

\(P_{3}(x) = 4 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	2	2			4
3	4	12			16
4	4	24	4		32
5	8	80	40		128
6	8	120	120	8	256
7	16	336	560	112	1024

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-903b47bd41c92af1 · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-4e15cddc85973b47

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 ?

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:  1;
  1:  2;
  2:  2,    2;
  3:  4,   12;
  4:  4,   24,    4;
  5:  8,   80,   40;
  6:  8,  120,  120,    8;
  7: 16,  336,  560,  112;
  8: 16,  448, 1120,  448,  16;
  9: 32, 1152, 4032, 2688, 288;

%F Formula

The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) - (4-8*x+4*x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 2 + 2*x, P(3,x) = 4 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 2 + 2*x;
P[3, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] - (4 - 8*x + 4*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] = 2 + 2*x; v[4] = 4 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] - (4 - 8*x + 4*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-da4f8499700ee86c
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-da4f8499700ee86c
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-da4f8499700ee86c

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-da4f8499700ee86c

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-07f7d31230240b98
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (10)

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

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