A209414 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	6	18	9													33
3	18	84	108	27												237
4	54	342	720	540	81											1737
5	162	1296	3798	4860	2430	243										12789
6	486	4698	17658	32130	28350	10206	729									94257
7	1458	16524	75816	179280	229068	149688	40824	2187								694845
8	4374	56862	308124	897480	1507572	1449252	734832	157464	6561							5122521
9	13122	192456	1202850	4165020	8687412	11092464	8389332	3411720	590490	19683						37764549
10	39366	642978	4553334	18276030	45586800	72801828	73806876	45323388	15155910	2165130	59049					278410689
11	118098	2125764	16822404	76807440	223095870	428378868	546767496	453811248	231668910	64953900	7794468	177147				2052521613
12	354294	6967782	60938568	311909940	1034324154	2322467154	3581889012	3765401640	2617642170	1131496938	270208224	27634932	531441			15131766249
13	1062882	22674816	217182222	1231762140	4592608146	11812791816	21387470526	27291826692	24162995142	14321035872	5319724410	1096185636	96722262	1594323		111555636885
14	3188646	73338858	763503570	4752854010	19684220346	57083862402	118762843626	178552431810	192706739550	146199537198	74932118118	24212806254	4352501790	334807830	4782969	822419536977

Technical details

Node id: n-3121cc0e9141d5d1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bf9e494f8c7b5418

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:     1;
  1:     2,      3;
  2:     6,     18,       9;
  3:    18,     84,     108,      27;
  4:    54,    342,     720,     540,      81;
  5:   162,   1296,    3798,    4860,    2430,      243;
  6:   486,   4698,   17658,   32130,   28350,    10206,     729;
  7:  1458,  16524,   75816,  179280,  229068,   149688,   40824,    2187;
  8:  4374,  56862,  308124,  897480, 1507572,  1449252,  734832,  157464,   6561;
  9: 13122, 192456, 1202850, 4165020, 8687412, 11092464, 8389332, 3411720, 590490, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(3*x+9*x^2)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(3*x + 9*x^2)*P[n - 2, x] + (3 + 6*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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*x + 9*x^2)*v[i - 2] + (3 + 6*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 coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.?
  - 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