Real-rooted atlas

A038330 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1872 + 1824\,x + 592\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
018721824592644352
1224642937614400313625669632
2269568442368290304952321561610241114112
3323481663866885253120230400056832074752409617825792
43881779289579520885841924866048016035840317030434816016384285212672
546581350412302254081421328384938262528387072000102187008168591361589248655364563402752
655897620481662595891221976842240169444638728397914112277453209661105766486507520714342426214473014444032
767077144576221870555136330225942528291240935424168552824832668860416001843082035234823208964317511683171942410485761168231104512
880492573491229307552399364850193530880481579499520031875976396801476843798528488714010624115511132160191102976002107637760139460608419430418691697672192
9965910881894438388765818880699253433303047719031406592057514351656960304725161410561177194332160033409896284166913680998401017328435201010407833660817408016777216299067162755072
10115909305827328499301625102336992659183239168120598514211225699893347614720059572760032051226315338442342487179648827392216603757117443986266521600528180314112477144023042634022912671088644785074604081152
111390911669927936645525672453734413909116699279360184424584383037441681114228221542411144465108434944554075101436313620987693236224006086431038504961344767011061762228322985574426852940840962224658841601134139801626843545676561193665298432
121669094003913523283026727374159872192730427289501696276945968056762368275503541139800064200978150430081024111066872606097408473482359409213441569879454069555240482928286760968053055626936321213564484321281341076694630410259603128324858681753610737418241224979098644774912
132002912804696227841063084488646459392264487203697065984040942733258391552004413826365904650240351375196972017254421367150729934929921012446321715445760377778478252032000111374692106895360258568380670279684677499631960064646354997084160659545915392004686883061760207232172032429496729619599665578316398592
14240349536563547340813558178985636003840359908023982337556485971076805795250176069343009694212423680598203291002606714883969558875480260608020696216152559321088858312702588616704028476102182908723207557808252319170561595573478516326402646625849285017633768750868070403200609628979202123431831142488046829568017179869184313594649253062377472

Computed in your browser ?

Technical details
Node id
n-0dc20949b68163c6 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-3aa9003708970881

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:          1872,           1824,            592,             64;
  1:         22464,          29376,          14400,           3136,            256;
  2:        269568,         442368,         290304,          95232,          15616,           1024;
  3:       3234816,        6386688,        5253120,        2304000,         568320,          74752,           4096;
  4:      38817792,       89579520,       88584192,       48660480,       16035840,        3170304,         348160,         16384;
  5:     465813504,     1230225408,     1421328384,      938262528,      387072000,      102187008,       16859136,       1589248,        65536;
  6:    5589762048,    16625958912,    21976842240,    16944463872,     8397914112,     2774532096,      611057664,      86507520,      7143424,       262144;
  7:   67077144576,   221870555136,   330225942528,   291240935424,   168552824832,    66886041600,    18430820352,    3482320896,    431751168,     31719424,     1048576;
  8:  804925734912,  2930755239936,  4850193530880,  4815794995200,  3187597639680,  1476843798528,   488714010624,  115511132160,  19110297600,   2107637760,   139460608,   4194304;
  9: 9659108818944, 38388765818880, 69925343330304, 77190314065920, 57514351656960, 30472516141056, 11771943321600, 3340989628416, 691368099840, 101732843520, 10104078336, 608174080, 16777216;
%F Formula
The row polynomials satisfy P(n,x) = (12+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 1824*x + 592*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 1872 + 1824*x + 592*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[(12 + 4*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] = 1872 + 1824*x + 592*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 4*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-60ab5e907432f9de

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.