A152842 → 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

\(P_{n}(x) = \left(2 + 6\,x\right)\,P_{n-2}(x) + \left(-1 + 10\,x - 9\,x^{2}\right)\,P_{n-4}(x)\)

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

\(P_{1}(x) = 3\)

\(P_{2}(x) = 4\)

\(P_{3}(x) = 7 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	3							3
2	4							4
3	7	9						16
4	8	24						32
5	11	90	27					128
6	12	136	108					256
7	15	307	621	81				1024
8	16	400	1200	432				2048
9	19	724	3858	3348	243			8192
10	20	880	5944	7920	1620			16384
11	23	1405	14374	33210	15795	729		65536
12	24	1640	19824	59472	44280	5832		131072
13	27	2414	40389	181380	229365	68526	2187	524288
14	28	2744	52164	281488	469476	222264	20412	1048576

Computed in your browser ?

Technical details

Node id: n-4371cb17de409031 · revision 1 · JSON
Definition source: derived:veronese_section_2_1
Fingerprint: f6-17702105e2bcc689

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 -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:  0;
  1:  3;
  2:  4;
  3:  7,   9;
  4:  8,  24;
  5: 11,  90,   27;
  6: 12, 136,  108;
  7: 15, 307,  621,   81;
  8: 16, 400, 1200,  432;
  9: 19, 724, 3858, 3348, 243;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x)*P(n-2,x) - (1-10*x+9*x^2)*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 4, P(3,x) = 7 + 9*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 4;
P[3, x_] := 7 + 9*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*x)*P[n - 2, x] - (1 - 10*x + 9*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] = 0; v[2] = 3; v[3] = 4; v[4] = 7 + 9*x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 6*x)*v[i - 2] - (1 - 10*x + 9*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-5ca24d5300044aef
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-5ca24d5300044aef
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-5ca24d5300044aef

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-5ca24d5300044aef

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 root: theorem check
    
    Parent-row provers of polytool 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