A272867 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 18\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	18	9													33
2	36	216	432	324	81											1089
3	216	1944	6804	11664	10206	4374	729									35937
4	1296	15552	77760	209952	332424	314928	174960	52488	6561							1185921
5	7776	116640	758160	2799360	6473520	9762768	9710280	6298560	2558790	590490	59049					39135393
6	46656	839808	6718464	31492800	96053040	200294208	292253184	300441312	216119340	106288200	34012224	6377292	531441			1291467969
7	279936	5878656	55847232	317447424	1203654816	3214155168	6223292208	8865853056	9334938312	7231849128	4062335004	1607077584	424089918	66961566	4782969	42618442977

Show all 15 stored rows

Generated b-files: A009977 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-c4fd821a14172d5f · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-3c4a6b4bc3495597

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): no

OEIS matches ?

A009977: 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:        6,        18,          9;
  2:       36,       216,        432,         324,           81;
  3:      216,      1944,       6804,       11664,        10206,         4374,           729;
  4:     1296,     15552,      77760,      209952,       332424,       314928,        174960,         52488,          6561;
  5:     7776,    116640,     758160,     2799360,      6473520,      9762768,       9710280,       6298560,       2558790,        590490,         59049;
  6:    46656,    839808,    6718464,    31492800,     96053040,    200294208,     292253184,     300441312,     216119340,     106288200,      34012224,       6377292,        531441;
  7:   279936,   5878656,   55847232,   317447424,   1203654816,   3214155168,    6223292208,    8865853056,    9334938312,    7231849128,    4062335004,    1607077584,     424089918,      66961566,       4782969;
  8:  1679616,  40310784,  443418624,  2962842624,  13438607616,  43807744512,  106027439616,  194141774592,  271604614752,  291212661888,  238561739136,  147851137728,   68032951056,   22499086176,    5050815264,    688747536,    43046721;
  9: 10077696, 272097792, 3401222400, 26121388032, 137953580544, 531406987776, 1545651507456, 3467614261248, 6078426587712, 8383435008192, 9117639881568, 7802132087808, 5216573837664, 2690247875616, 1047585002256, 297538935552, 58113073350, 6973568802, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (6+18*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[(6 + 18*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] = (6 + 18*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-0f2bf5edb3121755
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-0f2bf5edb3121755
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-0f2bf5edb3121755

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-0f2bf5edb3121755

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

A272867 → basis A013609