A308737 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-8 - 24\,x - 18\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 9\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	9							13
2	16	66	63						145
3	64	372	684	405					1525
4	256	1896	5076	5886	2511				15625
5	1024	9168	31968	54648	46008	15309			158125
6	4096	42912	183600	412560	515160	339714	92583		1590625
7	16384	196416	993600	2758320	4548960	4464396	2417364	557685	15953125

All 15 rows

Technical details

Node id: n-6f2154a01224373b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-527aea5dab844fde

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      4,       9;
  2:     16,      66,       63;
  3:     64,     372,      684,      405;
  4:    256,    1896,     5076,     5886,      2511;
  5:   1024,    9168,    31968,    54648,     46008,     15309;
  6:   4096,   42912,   183600,   412560,    515160,    339714,     92583;
  7:  16384,  196416,   993600,  2758320,   4548960,   4464396,   2417364,    557685;
  8:  65536,  884352,  5156928, 17013024,  34791120,  45219384,  36517068,  16765542,   3352671;
  9: 262144, 3931392, 25943040, 99042048, 241387776, 389896416, 417711168, 286409520, 114108912, 20135709;

%F Formula

The row polynomials satisfy P(n,x) = -(8+24*x+18*x^2)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 24*x + 18*x^2)*P[n - 2, x] + (6 + 9*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] = 1; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 24*x + 18*x^2)*v[i - 2] + (6 + 9*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 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - 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