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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	65537	699072	3184768	8044544	12211200	11239424	5963776	1572864	131072							43112257
9	262145	3145752	16427952	48779904	90176256	106592256	79306752	34799616	7667712	524288						387682633
10	1048577	13981040	82332960	281003520	612606720	885321728	850206720	526909440	195624960	36700160	2097152					3487832977
11	4194305	61516488	403274960	1555382400	3907031040	6677794816	7868084224	6322618368	3337027584	1066926080	173015040	8388608				31385253913
12	16777217	268435488	1938700992	8338073600	23714403840	46806171648	65395097600	64681672704	44449726464	20323500032	5674893312	805306368	33554432			282446313697
13	67108865	1163220344	9176516336	43541783168	138309966080	309636294656	500595310592	588544606208	499302662144	298393272320	119807148032	29553065984	3707764736	134217728		2541932937193
14	268435457	5010795216	42870137184	222454568448	780598086912	1954596876288	3590157312000	4893538516992	4945842929664	3658991271936	1927110721536	687026995200	151145938944	16911433728	536870912	22877060890417

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