Real-rooted atlas

A038226 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(3 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 264\,x + 640\,x^{2} + 512\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
0362646405121452
1108108040326656409615972
2324410420736522246553632768175692
397214904950403225606144006225922621441932612
42916524884043521728000442368067829765767168209715221258732
58748180792163296084188162709504055738368715653125242880016777216233846052
6262446123606345216383201281486356483839754246606028807298088964697620481342177282572306572
77873220470322393452816572211275246796823410114565053612032747424972872477573124160749568107374182428295372292
823619667709528817984068864256035831808001304277811233888927744628516454408153726976070464307200365072220168589934592311249095212
97085882220242431870713627713664001625868288067793780736206009008128459666358272747424972800863691079680673236123648317827579904687194767363423740047332
1021257647227597611337408001086375628870946979840333450805248116037727027230270711398405919605784576857047302144089292370083846339371728896274877906944054975581388837661140520652
11637729223383404039794302084166119526429975098982415679282544646148738252800183642315816964197538647244873068265340928953514951966729045201125376058961311039488236394999971844398046511104414272545727172
12191318767525204561380896294415681902745612325425315847101792681984309896407941121042826007674882728400120709125550078878023688706006083174401034167995334656900500023148544542608988307456202310139510784351843720888324556998002998892
1357395628241061637647447052480580928785920495217981440031165718298624149783263838208560764928655360165278084235264038477437599744007051864927371264100673088525434881097484403212288088318271501107204947802324992000172403423235276828147497671065650126978032987812
141721868847691014152161626088448212236277760019503969730560133114593411072698675537903616288056089667174494444619563008002476547801874432051937544861908992866168459766005761134630029167165441142942337073152008549802417586176044754521296994304146366987889541122251799813685248551396758362865932

Computed in your browser ?

Technical details
Node id
n-18347f928d4972a4 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-07069fdd2e8aedf5

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:     36,      264,       640,        512;
  1:    108,     1080,      4032,       6656,        4096;
  2:    324,     4104,     20736,      52224,       65536,       32768;
  3:    972,    14904,     95040,     322560,      614400,      622592,       262144;
  4:   2916,    52488,    404352,    1728000,     4423680,     6782976,      5767168,      2097152;
  5:   8748,   180792,   1632960,    8418816,    27095040,    55738368,     71565312,     52428800,     16777216;
  6:  26244,   612360,   6345216,   38320128,   148635648,   383975424,    660602880,    729808896,    469762048,    134217728;
  7:  78732,  2047032,  23934528,  165722112,   752467968,  2341011456,   5053612032,   7474249728,   7247757312,   4160749568,   1073741824;
  8: 236196,  6770952,  88179840,  688642560,  3583180800, 13042778112,  33888927744,  62851645440,  81537269760,  70464307200,  36507222016,   8589934592;
  9: 708588, 22202424, 318707136, 2771366400, 16258682880, 67793780736, 206009008128, 459666358272, 747424972800, 863691079680, 673236123648, 317827579904, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (3+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 264*x + 640*x^2 + 512*x^3.
%t Mathematica
P[0, x_] := 36 + 264*x + 640*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[(3 + 8*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] = 36 + 264*x + 640*x^2 + 512*x^3; for(n = 1, N, my(i = n + 1); v[i] = (3 + 8*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-c18c7d7bde35dd59

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.