A086645 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	3							5
3	8	24	9						41
4	32	144	162	27					365
5	128	768	1440	864	81				3281
6	512	3840	10080	10800	4050	243			29525
7	2048	18432	62208	96768	68040	17496	729		265721
8	8192	86016	354816	725760	762048	381024	71442	2187	2391485

Show all 15 stored rows

Generated b-files: A083884 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-831045384d9c95c9 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-38d9a649e263b473

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 ?

A083884: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      2,       3;
   3:      8,      24,       9;
   4:     32,     144,     162,       27;
   5:    128,     768,    1440,      864,       81;
   6:    512,    3840,   10080,    10800,     4050,      243;
   7:   2048,   18432,   62208,    96768,    68040,    17496,      729;
   8:   8192,   86016,  354816,   725760,   762048,   381024,    71442,    2187;
   9:  32768,  393216, 1916928,  4866048,  6842880,  5225472,  1959552,  279936,    6561;
  10: 131072, 1769472, 9953280, 30191616, 53374464, 55427328, 32332608, 9447840, 1062882, 19683;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = -9*x^2*v[i - 2] + (4 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-5c9b4fd96a349dee
Coefficients of one sign. Every nonzero row, n >= 1: 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-5c9b4fd96a349dee
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-5c9b4fd96a349dee
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-5c9b4fd96a349dee

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

basis_map A013610 e-5c9b4fd96a349dee

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-1375f2f1a11f9809 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-a5e3cd5a8b6d6a1a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n, k) = binomial(2n, 2k).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-1375f2f1a11f9809 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[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