A119808 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,x\right)\,P_{n-2}(x) + \left(3\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	1							4
2	9	5							14
3	27	21	2						50
4	81	81	16						178
5	243	297	90	4					634
6	729	1053	432	44					2258
7	2187	3645	1890	312	8				8042
8	6561	12393	7776	1800	112				28642
9	19683	41553	30618	9180	960	16			102010
10	59049	137781	116640	43092	6480	272			363314
11	177147	452709	433026	190512	37800	2736	32		1293962
12	531441	1476225	1574640	804816	199584	21168	640		4608514
13	1594323	4782969	5629338	3280500	979776	139104	7392	64	16413466
14	4782969	15411789	19840464	12990780	4548960	816480	64512	1472	58457426

Technical details

Node id: n-4b3a068fa525bce8 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-e8cd0fa6633bf52f

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     3,     1;
  2:     9,     5;
  3:    27,    21,     2;
  4:    81,    81,    16;
  5:   243,   297,    90,    4;
  6:   729,  1053,   432,   44;
  7:  2187,  3645,  1890,  312,   8;
  8:  6561, 12393,  7776, 1800, 112;
  9: 19683, 41553, 30618, 9180, 960, 16;

%F Formula

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

%t Mathematica

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

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceilin…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Candidate equalities ?

A119808 → x ↦ x+1 15 rows; no all-index equality certificate was found

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003