Real-rooted atlas

A038326 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0121224
1132276144552
2145246204896172812696
315972682441092967776020736292008
41756929423482021184216691211612162488326716184
5193261212474132335412004809024038776320166717442985984154472232
6212587321604067965186427849314870401003622400648705024232906752358318083552861336
72338460522019579540762995217616470070848222176908801917922406410346434560318903091242998169681715810728
82572306572250215275641081644284162727302054404420354498564775837552643439614689281592365547524299816960051597803521879463646744
928295372292306104482068149006704334442980054008328135152413696105578467061769514581221376587913972940823838185226245727356190726191736422443227663875112
103112490952123706693770252200639912616006515886392928014106274136064021375814273228823135455390924817884551168000096771680501760349059140812807553918435328743008370688994236269127576
1134237400473324450862061531626518422912062495751539836128023335965221184004044092466382848510999780578918447435552753909763210634625679360154522522091520050196407176396898820113301504891610044825622867434189934248
1237661140520652530679707336460345112996771065613714880131421568371597465236377607248817539563212810473908546027520011349908169877094492239644187164672555250929382195202406430744038604871105901074841601283918464548864106993205379072525950986368487704
13414272545727172628941046694888444330586132854736192277241058165120573335773337074176124328688763560652820219880448106127362505358924209782784237662506644406272017176517525663907849310084971028807683669881804669583369945018439984742416583946833756160128391846454886412096872686475217192
1445569980029988927415478568516378856310937306478870426470166852340731528614020399405797376205561850440365619203716131114454401843251822804704034963456562071828214020833284741367007555905126430852914498928377856152089719503711109125497810194001821696137582562796948684821313046511511142415407021574586368278228071788929995416

Computed in your browser ?

Technical details
Node id
n-00954e931f2325b9 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-c0bbc5ef1b77e180

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,           12;
  1:         132,          276,           144;
  2:        1452,         4620,          4896,          1728;
  3:       15972,        68244,        109296,         77760,         20736;
  4:      175692,       942348,       2021184,       2166912,       1161216,         248832;
  5:     1932612,     12474132,      33541200,      48090240,      38776320,       16671744,       2985984;
  6:    21258732,    160406796,     518642784,     931487040,    1003622400,      648705024,     232906752,      35831808;
  7:   233846052,   2019579540,    7629952176,   16470070848,   22217690880,    19179224064,   10346434560,    3189030912,     429981696;
  8:  2572306572,  25021527564,  108164428416,  272730205440,  442035449856,   477583755264,  343961468928,  159236554752,   42998169600,   5159780352;
  9: 28295372292, 306104482068, 1490067043344, 4298005400832, 8135152413696, 10557846706176, 9514581221376, 5879139729408, 2383818522624, 572735619072, 61917364224;
%F Formula
The row polynomials satisfy P(n,x) = (11+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 12*x.
%t Mathematica
P[0, x_] := 12 + 12*x;
P[n_, x_] := P[n, x] = Expand[(11 + 12*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 + 12*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 12*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-4797824a301b09a2

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.