A208509 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	6	2							8
3	12	10							22
4	24	32	4						60
5	48	88	28						164
6	96	224	120	8					448
7	192	544	416	72					1224
8	384	1280	1280	384	16				3344
9	768	2944	3648	1600	176				9136
10	1536	6656	9856	5760	1120	32			24960
11	3072	14848	25600	18816	5440	416			68192
12	6144	32768	64512	57344	22400	3072	64		186304
13	12288	71680	158720	165888	82432	17024	960		508992
14	24576	155648	382976	460800	279552	78848	8064	128	1390592

Technical details

Node id: n-6298766c9be8b1ca · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-452b40a247d9e038

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,    2;
  3:  12,   10;
  4:  24,   32,    4;
  5:  48,   88,   28;
  6:  96,  224,  120,    8;
  7: 192,  544,  416,   72;
  8: 384, 1280, 1280,  384,  16;
  9: 768, 2944, 3648, 1600, 176;

%F Formula

The row polynomials satisfy P(n,x) = 2*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[2*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] = 2*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