Real-rooted atlas

A038321 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(11 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1452 + 2695\,x + 1666\,x^{2} + 343\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
01452269516663436156
1159723980937191154352401110808
2175692549703687764430122134456168071994544
3193261272765771141332595456904489870112606911764935901792
42125873293570631176482614184895865116208400438158499176622823543646232256
52338460521178088065259630317132692328132572563455129543313940765378573295327576480111632180608
62572306572145958910793680595133654135683140511828276963225770871413552223608365982509257648010040353607209379250944
7282953722921785609478735070367022498531341738929419608866367131145897263748784206861351236412683196005674444792503772824752493768826516992
83112490952122162238032647682733035985012933732828555163335089702401443798669343891154647556284110508998750129742611306027299215135534461980378197732674367838877305856
93423740047332259633620256019023630018687919006237363305527020472847252527315241641949820133601916597410902385227564643045250234910120848964563252290026843643262984456819138412872011221099791505408
103766114052065230956316261293511743428362348762722340211271758430268862862916048961096799221534126763125762200260861450919392412366647185135324342506126639451097845704743481892301693051019931453569449688901040721979796247097344
1141427254572717236688227723868491508471333687418138166142177633470663859563938230668397602687954780379667162142839271575821014814685643186361346800632013433409768898119424738456386544397664517955604615438571743995415017796613085678223072849395636332447752192
12455699800299889243256958316345543191613606112323934525420557312087437997408515575488016138843799043178729514641709717280666021191073251296029101753574458518349468370812801733923549140755098584806706404747889221746078534759148731816124589578482361311258503009293447475615099437121453984059539456
135012697803298781250772552750079321724105483754499820757120921373219229345146494375725149802352225467750377807635725824946622031243687233510025663527863251662683180277404785593539620287016053154143995696570338609143050836823756761937720587309127222573448167674182455995970582478756584031838725991133232930569601128186171713071710208
1455139675836286593259358696487396400713007011082245535534495203973733561397320210990262910199388000347347515549163701572439857155368790215056437635654584099355384346352167794984030680219281404935135993450111061686863184961904447435619266056140721416282318166696780834483709560549125806236939377919404007896144303337197662477909470849882326305139872072307351090835290783744

Computed in your browser ?

Technical details
Node id
n-e22a7f0db816391e · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-0cc4706afbac7a71

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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:          1452,           2695,           1666,             343;
  1:         15972,          39809,          37191,           15435,            2401;
  2:        175692,         549703,         687764,          430122,          134456,           16807;
  3:       1932612,        7276577,       11413325,         9545690,         4489870,         1126069,          117649;
  4:      21258732,       93570631,      176482614,       184895865,       116208400,        43815849,         9176622,          823543;
  5:     233846052,     1178088065,     2596303171,      3269232813,      2572563455,      1295433139,       407653785,        73295327,        5764801;
  6:    2572306572,    14595891079,    36805951336,     54135683140,     51182827696,     32257708714,     13552223608,      3659825092,      576480100,       40353607;
  7:   28295372292,   178560947873,   507036702249,    853134173892,    941960886636,    713114589726,    374878420686,    135123641268,    31960056744,     4479250377,     282475249;
  8:  311249095212,  2162238032647,  6827330359850,  12933732828555,  16333508970240,  14437986693438,   9115464755628,   4110508998750,  1297426113060,   272992151355,   34461980378,   1977326743;
  9: 3423740047332, 25963362025601, 90236300186879, 190062373633055, 270204728472525, 273152416419498, 201336019165974, 109023852275646, 43045250234910, 12084896456325, 2290026843643, 262984456819, 13841287201;
%F Formula
The row polynomials satisfy P(n,x) = (11+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 1452 + 2695*x + 1666*x^2 + 343*x^3.
%t Mathematica
P[0, x_] := 1452 + 2695*x + 1666*x^2 + 343*x^3;
P[n_, x_] := P[n, x] = Expand[(11 + 7*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] = 1452 + 2695*x + 1666*x^2 + 343*x^3; for(n = 1, N, my(i = n + 1); v[i] = (11 + 7*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-57fae4cb2b7c6f75

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.