Real-rooted atlas

A038284 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 1248\,x + 900\,x^{2} + 216\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
057612489002162940
1460813440146887128129641160
236864135168198144145152531367776576240
32949121302528239616023500801296000381024466568067360
4235929612189696269844483317760024468480108241922659392279936112943040
518874368111673344289013760427327488394813440233404416862202881819584016796161581202560
61509949441006632960298215014451527024645722472448423611596820901888006628884481226119681007769622136835840
712079595528959033344298969989125911452057676695994368682237624324213820620817844240384495822643281629337660466176309915701760
896636764167892002406429293019136065229815808096825507840010059660656647464482242563955831603201467312537603627970560053814896643627970564338819824640
9773094113286893422510082816961675264697596641280011659829575680138572589957121200738218803276433546280963547348992000117062516736026073015091235191314432217678233660743477544960
106184752906245978594476032266717469081607270950135398413513443508224018081704941977617920261147852813319113015296074238919704576306490952908809109592211456184591142092822856214528013060694016850408685629440
1149478023249925153960755200024924554212147274170649228083215173324887818242257343005851648251852318834688021407447100948481393058138554368690626280554496256771309436928694248446361601290396568780814758584238087836416409611905721598812160
12395824185999364420036743659522303201982283776742912519097548816588898863939584271627389795041283369224354188492832237096810840064239889333690040321388335907576217661979281588224002096026613710848519780793319424892306615173129482063855616470184984576166680102383370240
1331665934879948837735239065272322107763790446592073252213421506560177285942057369600316835305019670528432514382212104192460050235738030080385334047817072640255000472820121600132883579725152256539557818626211841673440602882048038325300520550406112404799488006065386301030428211099074562333521433367183360
14253327479039590432088147345014784191262246674890752712483534798848000185780081698799616035983980925015818245361126887814856704627548817917686579258429737969647616004352008069463408640259307147472194764812289477332518830084576099394062909441310666765893632002788510415192064041526737837752323864920573214721692665944473632669300067140567040

Computed in your browser ?

Technical details
Node id
n-5a6ab0e4ba52fc63 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-797198ab3845c969

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:         576,         1248,           900,           216;
  1:        4608,        13440,         14688,          7128,           1296;
  2:       36864,       135168,        198144,        145152,          53136,           7776;
  3:      294912,      1302528,       2396160,       2350080,        1296000,         381024,          46656;
  4:     2359296,     12189696,      26984448,      33177600,       24468480,       10824192,        2659392,        279936;
  5:    18874368,    111673344,     289013760,     427327488,      394813440,      233404416,       86220288,      18195840,       1679616;
  6:   150994944,   1006632960,    2982150144,    5152702464,     5722472448,     4236115968,     2090188800,     662888448,     122611968,      10077696;
  7:  1207959552,   8959033344,   29896998912,   59114520576,    76695994368,    68223762432,    42138206208,   17844240384,    4958226432,     816293376,     60466176;
  8:  9663676416,  78920024064,  292930191360,  652298158080,   968255078400,  1005966065664,   746448224256,  395583160320,  146731253760,   36279705600,   5381489664,   362797056;
  9: 77309411328, 689342251008, 2816961675264, 6975966412800, 11659829575680, 13857258995712, 12007382188032, 7643354628096, 3547348992000, 1170625167360, 260730150912, 35191314432, 2176782336;
%F Formula
The row polynomials satisfy P(n,x) = (8+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1248*x + 900*x^2 + 216*x^3.
%t Mathematica
P[0, x_] := 576 + 1248*x + 900*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[(8 + 6*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] = 576 + 1248*x + 900*x^2 + 216*x^3; for(n = 1, N, my(i = n + 1); v[i] = (8 + 6*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-c0fedf7d6d9f66e7

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.