A034807 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	3	8							11
3	4	12							16
4	7	32	32						71
5	11	60	80						151
6	18	120	240	128					506
7	29	224	560	448					1261
8	47	416	1280	1536	512				3791
9	76	756	2736	4224	2304				10096
10	123	1360	5680	10880	8960	2048			29051
11	199	2420	11440	26048	28160	11264			79531
12	322	4272	22560	59648	80640	49152	8192		224786
13	521	7488	43680	131456	212992	173056	53248		622441
14	843	13048	83328	281344	532224	544768	258048	32768	1746371

Technical details

Node id: n-5f5f9434b2e29a17 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5ed469058d4aab7f

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,   8;
  3:  4,  12;
  4:  7,  32,   32;
  5: 11,  60,   80;
  6: 18, 120,  240,  128;
  7: 29, 224,  560,  448;
  8: 47, 416, 1280, 1536,  512;
  9: 76, 756, 2736, 4224, 2304;

%F Formula

The row polynomials satisfy P(n,x) = (1+4*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 + 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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 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