A200536 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 28\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)

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

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	28	32													66
2	36	336	1168	1792	1024											4356
3	216	3024	17568	54208	93696	86016	32768									287496
4	1296	24192	196992	913920	2642176	4874240	5603328	3670016	1048576							18974736
5	7776	181440	1900800	11773440	47746560	132471808	254648320	334888960	288358400	146800640	33554432					1252332576
6	46656	1306368	16733952	129669120	676961280	2508484608	6764990464	13378584576	19255787520	19671285760	13539213312	5637144576	1073741824			82653950016
7	279936	9144576	138475008	1288369152	8227989504	38155235328	132490272768	349962747904	706614788096	1085304471552	1248216481792	1042401984512	597537325056	210453397504	34359738368	5455160701056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-02d4e89e51dfb496 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e0420e98772e8ba9

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 ?

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:        6,        28,         32;
  2:       36,       336,       1168,         1792,         1024;
  3:      216,      3024,      17568,        54208,        93696,         86016,          32768;
  4:     1296,     24192,     196992,       913920,      2642176,       4874240,        5603328,         3670016,         1048576;
  5:     7776,    181440,    1900800,     11773440,     47746560,     132471808,      254648320,       334888960,       288358400,       146800640,         33554432;
  6:    46656,   1306368,   16733952,    129669120,    676961280,    2508484608,     6764990464,     13378584576,     19255787520,     19671285760,      13539213312,       5637144576,       1073741824;
  7:   279936,   9144576,  138475008,   1288369152,   8227989504,   38155235328,   132490272768,    349962747904,    706614788096,   1085304471552,    1248216481792,    1042401984512,     597537325056,     210453397504,      34359738368;
  8:  1679616,  62705664, 1095856128,  11900141568,  89873473536,  500542930944,  2126583889920,   7030471655424,  18278334398464,  37495848828928,   60489497313280,   75934216486912,   72715406934016,   51350628990976,   25220047962112,    7696581394432,    1099511627776;
  9: 10077696, 423263232, 8384643072, 104091402240, 907512201216, 5900519374848, 29650656559104, 117744552640512, 374573897220096, 961743549104128, 1997727451840512, 3349178386219008, 4498114417262592, 4774030095679488, 3916030921408512, 2395560959016960, 1029142883598336, 277076930199552, 35184372088832;

%F Formula

The row polynomials satisfy P(n,x) = (6+28*x+32*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 + 28*x + 32*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 + 28*x + 32*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-ecef41f482d88baf
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-ecef41f482d88baf
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-ecef41f482d88baf

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 A013611 e-ecef41f482d88baf

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-982ad459e6d62a8e root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.