A201730 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 + 2\,x\right)\,P_{n-2}(x) + \left(4\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2								2
3	6	2							8
4	20	12							32
5	68	56	4						128
6	232	240	40						512
7	792	984	264	8					2048
8	2704	3920	1456	112					8192
9	9232	15296	7264	960	16				32768
10	31520	58752	33984	6528	288				131072
11	107616	222880	152000	38720	3040	32			524288
12	367424	837056	657536	209792	24640	704			2097152
13	1254464	3117696	2771904	1065728	169920	8832	64		8388608
14	4283008	11531520	11446656	5158400	1049984	83200	1664		33554432
15	14623104	42399616	46478208	24045952	5991552	654976	24192	128	134217728

Technical details

Node id: n-53e4084123346ddd · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-30a6ae34b58580ed

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     2;
   3:     6,     2;
   4:    20,    12;
   5:    68,    56,     4;
   6:   232,   240,    40;
   7:   792,   984,   264,    8;
   8:  2704,  3920,  1456,  112;
   9:  9232, 15296,  7264,  960,  16;
  10: 31520, 58752, 33984, 6528, 288;

%F Formula

The row polynomials satisfy P(n,x) = -(2-2*x)*P(n-2,x) + 4*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(2 - 2*x)*P[n - 2, x] + 4*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(2 - 2*x)*v[i - 2] + 4*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA …?
  - 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