A274498 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 12\,x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	9	12							21
3	27	36							63
4	81	216	144						441
5	243	648	432						1323
6	729	2916	3888	1728					9261
7	2187	8748	11664	5184					27783
8	6561	34992	69984	62208	20736				194481
9	19683	104976	209952	186624	62208				583443
10	59049	393660	1049760	1399680	933120	248832			4084101
11	177147	1180980	3149280	4199040	2799360	746496			12252303
12	531441	4251528	14171760	25194240	25194240	13436928	2985984		85766121
13	1594323	12754584	42515280	75582720	75582720	40310784	8957952		257298363
14	4782969	44641044	178564176	396809280	529079040	423263232	188116992	35831808	1801088541

Computed in your browser ?

Technical details

Node id: n-612921ba5e07601f · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b047158a8b919e42

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): yes

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:     3;
  2:     9,     12;
  3:    27,     36;
  4:    81,    216,    144;
  5:   243,    648,    432;
  6:   729,   2916,   3888,   1728;
  7:  2187,   8748,  11664,   5184;
  8:  6561,  34992,  69984,  62208, 20736;
  9: 19683, 104976, 209952, 186624, 62208;

%F Formula

The row polynomials satisfy P(n,x) = (9+12*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(9 + 12*x)*P[n - 2, 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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (9 + 12*x)*v[i - 2]); 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-8f465536c6a75fc0
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-8f465536c6a75fc0
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-8f465536c6a75fc0

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 A013609 e-8f465536c6a75fc0

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-363ec3019ccc29a9 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.

A188440 → basis A013609