Real-rooted atlas

A038323 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(11 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
012921
113220781420
21452346527547298400
3159725118361479328056561168000
41756927067611136916914166367416590493360000
519326129355599188669252028807012269070395628353144167200000
6212587321203050972917365663929710953175524001539407434145239847829691344000000
72338460521514684655429184809969483111397029816255455131977318414430654256842414304672126880000000
82572306572187661456736084249098411505805542013986277905611333286379861217751672212555142364304672100387420489537600000000
928295372292229578361551838162711881181322102847625740130683962505426513282169339104257478477042164423865102122443003674279348678440110752000000000
1031124909521227800203276891128599508465027488895720165446331330086405072580926166641176140087852238729940212509688095028260262089960808542538769692231381059609215040000000000
113423740047332333814654614871491661288803514039518086836657383645245765259596820989560869094698243213666331881950244183214259915021101160227509432752826736113890741736809279972824295364814300800000000000
1237661140520652398009780502345194126060683724457859650554434741175757604849476017201783809705671186413069581398001515029856416089292343796617429724169084183895015135514573101745230031674045813040669853253264254186582832986016000000000000
134142725457271724717057850211663249359546997307898111696107141341018140702203243362629503780634321522135987043082688883933442504682902801223793086335662072012896934297853191352428360696247107154997957241964983150219049909074393989203390995228767924549611720320000000000000
144556998002998892556160892638728413167490223489436461116710164083124611272552989199949057648780790680672700656613914996184714218691750939256130768356270649183840800283560040542973437523173743605446550539464235299923238544117415057106676829632685852686482611379754754752352620589113209464934406400000000000000

Computed in your browser ?

Technical details
Node id
n-1438be315d9d27d7 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-9957a1cf35786f48

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 ?

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:          12,            9;
  1:         132,          207,           81;
  2:        1452,         3465,         2754,           729;
  3:       15972,        51183,        61479,         32805,          6561;
  4:      175692,       706761,      1136916,        914166,        367416,         59049;
  5:     1932612,      9355599,     18866925,      20288070,      12269070,       3956283,        531441;
  6:    21258732,    120305097,    291736566,     392971095,     317552400,     153940743,      41452398,      4782969;
  7:   233846052,   1514684655,   4291848099,    6948311139,    7029816255,    4551319773,    1841443065,    425684241,     43046721;
  8:  2572306572,  18766145673,  60842490984,  115058055420,  139862779056,  113332863798,   61217751672,  21255514236,   4304672100,   387420489;
  9: 28295372292, 229578361551, 838162711881, 1813221028476, 2574013068396, 2505426513282, 1693391042574, 784770421644, 238651021224, 43003674279, 3486784401;
%F Formula
The row polynomials satisfy P(n,x) = (11+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 9*x.
%t Mathematica
P[0, x_] := 12 + 9*x;
P[n_, x_] := P[n, x] = Expand[(11 + 9*x)*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] = 12 + 9*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 9*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 - 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-316947e1f7e184b6

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

No candidate equalities or related sequences.

References

  1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.