A208509 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	6	3							9
3	12	15							27
4	24	48	9						81
5	48	132	63						243
6	96	336	270	27					729
7	192	816	936	243					2187
8	384	1920	2880	1296	81				6561
9	768	4416	8208	5400	891				19683
10	1536	9984	22176	19440	5670	243			59049
11	3072	22272	57600	63504	27540	3159			177147
12	6144	49152	145152	193536	113400	23328	729		531441
13	12288	107520	357120	559872	417312	129276	10935		1594323
14	24576	233472	861696	1555200	1415232	598752	91854	2187	4782969

Technical details

Node id: n-bb7c767681c825f9 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-3e1c33a82a15280b

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   3;
  2:   6,    3;
  3:  12,   15;
  4:  24,   48,    9;
  5:  48,  132,   63;
  6:  96,  336,  270,   27;
  7: 192,  816,  936,  243;
  8: 384, 1920, 2880, 1296,  81;
  9: 768, 4416, 8208, 5400, 891;

%F Formula

The row polynomials satisfy P(n,x) = 3*x*P(n-2,x) + 2*P(n-1,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[3*x*P[n - 2, x] + 2*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; for(n = 2, N, my(i = n + 1); v[i] = 3*x*v[i - 2] + 2*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 of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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