A133080 → basis A013610

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

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

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

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	1	6	9													16
3	2	15	36	27												80
4	1	12	54	108	81											256
5	2	27	144	378	486	243										1280
6	1	18	135	540	1215	1458	729									4096
7	2	39	324	1485	4050	6561	5832	2187								20480
8	1	24	252	1512	5670	13608	20412	17496	6561							65536
9	2	51	576	3780	15876	44226	81648	96228	65610	19683						327680
10	1	30	405	3240	17010	61236	153090	262440	295245	196830	59049					1048576
11	2	63	900	7695	43740	173502	489888	984150	1377810	1279395	708588	177147				5242880
12	1	36	594	5940	40095	192456	673596	1732104	3247695	4330260	3897234	2125764	531441			16777216
13	2	75	1296	13662	98010	505197	1924560	5484996	11691702	18403605	20785248	15943230	7440174	1594323		83886080
14	1	42	819	9828	81081	486486	2189187	7505784	19702683	39405366	59108049	64481508	48361131	22320522	4782969	268435456

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Technical details

Node id: n-911b7f6688b0114d · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4a4307281c543b87

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 2,  3;
  2: 1,  6,   9;
  3: 2, 15,  36,   27;
  4: 1, 12,  54,  108,    81;
  5: 2, 27, 144,  378,   486,   243;
  6: 1, 18, 135,  540,  1215,  1458,   729;
  7: 2, 39, 324, 1485,  4050,  6561,  5832,  2187;
  8: 1, 24, 252, 1512,  5670, 13608, 20412, 17496,  6561;
  9: 2, 51, 576, 3780, 15876, 44226, 81648, 96228, 65610, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (1+6*x+9*x^2)*P(n-2,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[(1 + 6*x + 9*x^2)*P[n - 2, 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] = (1 + 6*x + 9*x^2)*v[i - 2]); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-e4e51bf785131387
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-e4e51bf785131387
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-e4e51bf785131387

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013610 e-e4e51bf785131387

Geometric basis map (elementary) polytool ?
- premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.