basis-map coefficients of A112227 → 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

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

\(P_{1}(x) = 1 + 6\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	1	6	9															16
2	1	9	27	27														64
3	1	12	54	108	81													256
4	1	15	90	270	405	243												1024
5	1	18	135	540	1215	1458	729											4096
6	1	21	189	945	2835	5103	5103	2187										16384
7	1	24	252	1512	5670	13608	20412	17496	6561									65536
8	1	27	324	2268	10206	30618	61236	78732	59049	19683								262144
9	1	30	405	3240	17010	61236	153090	262440	295245	196830	59049							1048576
10	1	33	495	4455	26730	112266	336798	721710	1082565	1082565	649539	177147						4194304
11	1	36	594	5940	40095	192456	673596	1732104	3247695	4330260	3897234	2125764	531441					16777216
12	1	39	702	7722	57915	312741	1250964	3752892	8444007	14073345	16888014	13817466	6908733	1594323				67108864
13	1	42	819	9828	81081	486486	2189187	7505784	19702683	39405366	59108049	64481508	48361131	22320522	4782969			268435456
14	1	45	945	12285	110565	729729	3648645	14073345	42220035	98513415	177324147	241805655	241805655	167403915	71744535	14348907		1073741824
15	1	48	1080	15120	147420	1061424	5837832	25019280	84440070	225173520	472864392	773778096	967222620	892820880	573956280	229582512	43046721	4294967296

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Technical details

Node id: n-b8d8f9de0e27e805 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-f9762973ec1089d7

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 1)
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 ?

A347478: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A000302: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A155543: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,  6,   9;
   2: 1,  9,  27,   27;
   3: 1, 12,  54,  108,    81;
   4: 1, 15,  90,  270,   405,    243;
   5: 1, 18, 135,  540,  1215,   1458,    729;
   6: 1, 21, 189,  945,  2835,   5103,   5103,   2187;
   7: 1, 24, 252, 1512,  5670,  13608,  20412,  17496,    6561;
   8: 1, 27, 324, 2268, 10206,  30618,  61236,  78732,   59049,   19683;
   9: 1, 30, 405, 3240, 17010,  61236, 153090, 262440,  295245,  196830,  59049;
  10: 1, 33, 495, 4455, 26730, 112266, 336798, 721710, 1082565, 1082565, 649539, 177147;

%F Formula

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

%t Mathematica

P[1, x_] := 1 + 6*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 6*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (1 + 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-33def10999a9e794
Coefficients of one sign. Every nonzero row, n >= 1: 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-33def10999a9e794
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-33def10999a9e794

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-33def10999a9e794

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of A scaled Hermite triangle.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-470643e70af29063 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-f9cce4eb6656235e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Related sequences ?

Same fingerprint (16)

Equal up to content, sign, reversal and a small index shift.