A391533 → 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

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

\(P_{0}(x) = 9 + 18\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	18	9															36
1	18	63	72	27														180
2	36	180	333	270	81													900
3	72	468	1206	1539	972	243												4500
4	144	1152	3816	6696	6561	3402	729											22500
5	288	2736	11088	24840	33210	26487	11664	2187										112500
6	576	6336	30384	82944	140940	152604	102789	39366	6561									562500
7	1152	14400	79776	257040	530712	728028	663390	387099	131220	19683								2812500
8	2304	32256	202752	753408	1832544	3048192	3510864	2764368	1423737	433026	59049							14062500
9	4608	71424	502272	2115072	5925312	11594016	16166304	16061328	11140578	5137263	1417176	177147						70312500
10	9216	156672	1218816	5736960	18195840	40963968	67114656	80621568	70465140	43696260	18246141	4605822	531441					351562500
11	18432	340992	2907648	15130368	53602560	136515456	257121216	362587104	382794984	298787940	167581062	63950067	14880348	1594323				1757812500
12	36864	737280	6838272	38983680	152596224	433838592	923788800	1496537856	1853351280	1745960832	1231525944	630643320	221610897	47829690	4782969			8789062500
13	73728	1585152	15888384	98482176	422143488	1325465856	3149093376	5764442112	8196316128	9051975504	7700934384	4955864472	2335151754	760492071	153055008	14348907		43945312500
14	147456	3391488	36532224	244629504	1139733504	3917362176	10274584320	20976164352	33685958592	42692899392	42557795280	33014532096	19537896924	8526439404	2587586229	487862838	43046721	219726562500

Computed in your browser ?

Technical details

Node id: n-b74e5c4b49294e5a · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f26e13ac80424634

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:    9,    18,      9;
  1:   18,    63,     72,      27;
  2:   36,   180,    333,     270,      81;
  3:   72,   468,   1206,    1539,     972,      243;
  4:  144,  1152,   3816,    6696,    6561,     3402,      729;
  5:  288,  2736,  11088,   24840,   33210,    26487,    11664,     2187;
  6:  576,  6336,  30384,   82944,  140940,   152604,   102789,    39366,     6561;
  7: 1152, 14400,  79776,  257040,  530712,   728028,   663390,   387099,   131220,   19683;
  8: 2304, 32256, 202752,  753408, 1832544,  3048192,  3510864,  2764368,  1423737,  433026,   59049;
  9: 4608, 71424, 502272, 2115072, 5925312, 11594016, 16166304, 16061328, 11140578, 5137263, 1417176, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 18*x + 9*x^2.

%t Mathematica

P[0, x_] := 9 + 18*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 3*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] = 9 + 18*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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)

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-68fe4bad60475721
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-68fe4bad60475721
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-68fe4bad60475721

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-68fe4bad60475721

Geometric basis map (elementary) polytool ?
- premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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