A103633 → 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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	2	9	9													20
3	2	15	36	27												80
4	4	36	117	162	81											400
5	4	48	225	513	567	243										1600
6	8	108	594	1701	2673	2187	729									8000
7	8	132	918	3483	7776	10206	7290	2187								32000
8	16	288	2232	9720	26001	43740	45198	26244	6561							160000
9	16	336	3096	16416	55161	121743	176418	161838	85293	19683						640000
10	32	720	7200	42120	159570	408969	718065	852930	656100	295245	59049					3200000
11	32	816	9360	63720	285930	887679	1944972	3007125	3214890	2263545	944784	177147				12800000
12	64	1728	21168	155520	763020	2633148	6552981	11849166	15451155	14171760	8680203	3188646	531441			64000000
13	64	1920	26352	219024	1229580	4922208	14452425	31508109	50998653	60525225	51195483	29229255	10097379	1594323		256000000
14	128	4032	58464	517104	3116232	13533156	43671474	106373493	196521633	274046409	283966641	212044959	107882523	33480783	4782969	1280000000

Computed in your browser ?

Technical details

Node id: n-8f297cd827c43952 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f25c978562657419

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:  1,   3;
  2:  2,   9,    9;
  3:  2,  15,   36,    27;
  4:  4,  36,  117,   162,    81;
  5:  4,  48,  225,   513,   567,    243;
  6:  8, 108,  594,  1701,  2673,   2187,    729;
  7:  8, 132,  918,  3483,  7776,  10206,   7290,   2187;
  8: 16, 288, 2232,  9720, 26001,  43740,  45198,  26244,  6561;
  9: 16, 336, 3096, 16416, 55161, 121743, 176418, 161838, 85293, 19683;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 9*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 9*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-473a0c8b41a1e681
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-473a0c8b41a1e681
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-473a0c8b41a1e681

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-473a0c8b41a1e681

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.