basis-map coefficients of A297191, shifted → 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

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

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

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	6	9													16
2	1	12	54	108	81											256
3	1	18	135	540	1215	1458	729									4096
4	1	24	252	1512	5670	13608	20412	17496	6561							65536
5	1	30	405	3240	17010	61236	153090	262440	295245	196830	59049					1048576
6	1	36	594	5940	40095	192456	673596	1732104	3247695	4330260	3897234	2125764	531441			16777216
7	1	42	819	9828	81081	486486	2189187	7505784	19702683	39405366	59108049	64481508	48361131	22320522	4782969	268435456

Show all 15 stored rows

Generated b-files: A001025 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-eb3f8d5fd252d0c9 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-12929af9fbcc0757

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A001025: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,  6,    9;
  2: 1, 12,   54,   108,     81;
  3: 1, 18,  135,   540,   1215,    1458,      729;
  4: 1, 24,  252,  1512,   5670,   13608,    20412,    17496,      6561;
  5: 1, 30,  405,  3240,  17010,   61236,   153090,   262440,    295245,    196830,      59049;
  6: 1, 36,  594,  5940,  40095,  192456,   673596,  1732104,   3247695,   4330260,    3897234,    2125764,     531441;
  7: 1, 42,  819,  9828,  81081,  486486,  2189187,  7505784,  19702683,  39405366,   59108049,   64481508,   48361131,    22320522,     4782969;
  8: 1, 48, 1080, 15120, 147420, 1061424,  5837832, 25019280,  84440070, 225173520,  472864392,  773778096,  967222620,   892820880,   573956280,   229582512,   43046721;
  9: 1, 54, 1377, 22032, 247860, 2082024, 13533156, 69599088, 287096238, 956987460, 2583866142, 5637526128, 9865670724, 13660159464, 14635885140, 11708708112, 6586148313, 2324522934, 387420489;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 6*x + 9*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + 6*x + 9*x^2)*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-4aebfd11077ddb65
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-4aebfd11077ddb65
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-4aebfd11077ddb65

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-4aebfd11077ddb65

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-1991a148b9538b93 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-12303ad0e0b9f8eb
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)

Related sequences ?

No candidate equalities or related sequences.