A034807 → basis A013610

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 2\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	3	6							9
3	4	9							13
4	7	24	18						49
5	11	45	45						101
6	18	90	135	54					297
7	29	168	315	189					701
8	47	312	720	648	162				1889
9	76	567	1539	1782	729				4693
10	123	1020	3195	4590	2835	486			12249
11	199	1815	6435	10989	8910	2673			31021
12	322	3204	12690	25164	25515	11664	1458		80017
13	521	5616	24570	55458	67392	41067	9477		204101
14	843	9786	46872	118692	168399	129276	45927	4374	524169

Technical details

Node id: n-9e175ec83e8af6f6 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9ec676f9f53aaaaa

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:  2;
  1:  1;
  2:  3,   6;
  3:  4,   9;
  4:  7,  24,   18;
  5: 11,  45,   45;
  6: 18,  90,  135,   54;
  7: 29, 168,  315,  189;
  8: 47, 312,  720,  648, 162;
  9: 76, 567, 1539, 1782, 729;

%F Formula

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

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 2, x] + 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] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 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 T(n,k) of coefficients of Lucas (or Cardan) polynomials.?
  - 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