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	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	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
8	1679616	62705664	1095856128	11900141568	89873473536	500542930944	2126583889920	7030471655424	18278334398464	37495848828928	60489497313280	75934216486912	72715406934016	51350628990976	25220047962112	7696581394432	1099511627776													360040606269696
9	10077696	423263232	8384643072	104091402240	907512201216	5900519374848	29650656559104	117744552640512	374573897220096	961743549104128	1997727451840512	3349178386219008	4498114417262592	4774030095679488	3916030921408512	2395560959016960	1029142883598336	277076930199552	35184372088832											23762680013799936
10	60466176	2821754880	62481715200	872862842880	8627941048320	64144382754816	372158872289280	1725502319493120	6493111867146240	20026356101283840	50901548797001728	106807232540180480	184692959776604160	261765092616437760	301108689559879680	276791194615283712	198563553639137280	107136413010493440	40901832553267200	9851624184872960	1125899906842624									1568336880910795776
11	362797056	18623582208	455834345472	7076961239040	78207220776960	654380256854016	4305090064416768	22826082589212672	99181820061941760	357181343111577600	1073926843366637568	2706930156798214144	5727609831288733696	10159824870729318400	15046249443470868480	18468273439094341632	18577012836978917376	15059916207647883264	9599264276065812480	4632726272936509440	1591459518322049024	346777171307528192	36028797018963968							103510234140112521216
12	2176782336	121899810816	3268075880448	55821083738112	681984938409984	6342546482528256	46655818643275776	278439185558274048	1371984114930941952	5650613663258640384	19618456909306134528	57741335534625619968	144525362365814800384	307953789518003306496	558036107642485604352	857219020915088556032	1110053715507582861312	1201500606732315918336	1073737650253860962304	778493076012194070528	446441549586260557824	194888770274830843904	60852638165030141952	12105675798371893248	1152921504606846976					6831675453247426400256
13	13060694016	792348770304	23091307020288	430333421027328	5759478403301376	58937131850268672	479349731399565312	3179959522802270208	17521188081802149888	81229291135482986496	319831415704868880384	1076584443892601978880	3111700190262202466304	7742155620458854088704	16595734348065079820288	30622846404056234065920	48519610323227516076032	65721516343889049419776	75606161786270663049216	73183632078615380754432	58836060233982548115456	38581474442454490939392	20108130983445782331392	8012948572205662339072	2293160872663018635264	419663427676892299264	36893488147419103232			450890579914330142416896
14	78364164096	5119792054272	161151549898752	3253912283381760	47345128033222656	528658855866925056	4710641389110558720	34387537835210047488	209505186534062555136	1079727717833030762496	4754086664640405700608	18014123619427396091904	59049170873121874378752	168031241254658056912896	415929169549628872327168	896166620024842970202112	1679620860391022204551168	2732809272043504236756992	3846466958689797876482048	4659162845491720431337472	4821555236765787167391744	4220774759721930477010944	3083683997776841979985920	1845712541128259589373952	881581716728001691910144	323140839311207070433280	85353084829054095327232	14462247353788288466944	1180591620717411303424	29758778274345789399515136

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.