A199478 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	2								4
1	4	6								10
2	8	18	8							34
3	16	46	32							94
4	32	114	120	32						298
5	64	270	368	160						862
6	128	626	1064	704	128					2650
7	256	1422	2880	2496	768					7822
8	512	3186	7512	8160	3840	512				23722
9	1024	7054	18960	24672	15360	3584				70654
10	2048	15474	46728	71040	55680	19968	2048			212986
11	4096	33678	112864	196224	185088	88576	16384			636910
12	8192	72818	268216	525216	580608	351232	100352	8192		1914826
13	16384	156558	628656	1369120	1735680	1268736	487424	73728		5736286
14	32768	334962	1456360	3492416	4997760	4293632	2093056	491520	32768	17225242

Technical details

Node id: n-5811f054b318ea57 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-221ad2ec859701ae

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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,    2;
  1:    4,    6;
  2:    8,   18,     8;
  3:   16,   46,    32;
  4:   32,  114,   120,    32;
  5:   64,  270,   368,   160;
  6:  128,  626,  1064,   704,   128;
  7:  256, 1422,  2880,  2496,   768;
  8:  512, 3186,  7512,  8160,  3840,  512;
  9: 1024, 7054, 18960, 24672, 15360, 3584;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 4 + 6*x.

%t Mathematica

P[0, x_] := 2 + 2*x;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*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 + 2*x; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*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 Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A209599 → basis A013611

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