A122076 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 16\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(5 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	5	8							13
2	17	48	32						97
3	65	264	336	128					793
4	257	1376	2624	2048	512				6817
5	1025	6840	17520	21120	11520	2048			60073
6	4097	32784	105696	173568	149760	61440	8192		535537
7	16385	152936	594832	1239168	1473024	974848	315392	32768	4799353

All 15 rows

Technical details

Node id: n-24993e39361130a7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-fc73a6df2d1bffc9

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:      2;
  1:      5,       8;
  2:     17,      48,       32;
  3:     65,     264,      336,      128;
  4:    257,    1376,     2624,     2048,      512;
  5:   1025,    6840,    17520,    21120,    11520,      2048;
  6:   4097,   32784,   105696,   173568,   149760,     61440,     8192;
  7:  16385,  152936,   594832,  1239168,  1473024,    974848,   315392,    32768;
  8:  65537,  699072,  3184768,  8044544, 12211200,  11239424,  5963776,  1572864,  131072;
  9: 262145, 3145752, 16427952, 48779904, 90176256, 106592256, 79306752, 34799616, 7667712, 524288;

%F Formula

The row polynomials satisfy P(n,x) = -(4+16*x+16*x^2)*P(n-2,x) + (5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 8*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 16*x + 16*x^2)*P[n - 2, x] + (5 + 8*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] = 5 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 16*x + 16*x^2)*v[i - 2] + (5 + 8*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 (−∞, 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 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.?
  - 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