A236076 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

\(P_{1}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	4	7	3													14
3	8	21	18	5												52
4	16	55	70	39	8											188
5	32	137	232	194	80	13										688
6	64	327	690	769	477	156	21									2504
7	128	761	1924	2680	2220	1093	296	34								9136
8	256	1735	5110	8539	8851	5825	2376	549	55							33296
9	512	3897	13104	25542	31794	26202	14292	4974	1002	89						121408
10	1024	8647	32682	72789	105765	104738	71572	33317	10109	1805	144					442592
11	2048	19001	79740	199764	331872	383831	315294	183915	74644	20072	3218	233				1613632
12	4096	41415	191070	531775	994215	1315263	1262248	885297	450300	161970	39102	5688	377			5882816
13	8192	89657	451016	1380594	2868766	4272520	4691464	3845088	2351924	1060261	342368	74982	9982	610		21447424
14	16384	192967	1051154	3509785	8024185	13286232	16428484	15409153	11015751	5972322	2417043	708002	141884	17411	987	78191744

Technical details

Node id: n-fb2c8b0ba28661fd · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-f9ff5861407dbab6

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 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:   2,    2;
  2:   4,    7,     3;
  3:   8,   21,    18,     5;
  4:  16,   55,    70,    39,     8;
  5:  32,  137,   232,   194,    80,    13;
  6:  64,  327,   690,   769,   477,   156,    21;
  7: 128,  761,  1924,  2680,  2220,  1093,   296,   34;
  8: 256, 1735,  5110,  8539,  8851,  5825,  2376,  549,   55;
  9: 512, 3897, 13104, 25542, 31794, 26202, 14292, 4974, 1002, 89;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x + x^2)*P[n - 2, x] + (1 + 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] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 3*x + x^2)*v[i - 2] + (1 + 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 - 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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 A skewed version of triangular array A122075.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from A skewed version of triangular array A122075.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

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