Real-rooted atlas

A038335 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(12 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 13 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
013922
115622581462
21872410429977299702
3224646609672900357216561203742
426956899532814696641084752400221590494278582
5323481614370048265939202624400014565420431057753144189850222
6388177922015539204484574725542732804109810401828157044517248547829691886854662
746581350427680071687195474944106873966089920232000589261780821874111564639479934304672139623947902
855897620483740840755211125776384019300803379221522935347215999350169679282494144252540763204692092589387420489832102905942
96707714457649920874905616717688340483317416280064431982454579238569862016002391331444992101659136313628359179794846877879169348678440117474161024782
10804925734912659419928985624554104750080548549148672008169464107008085162255331328634088531543043372107936256012552423843600311486073156046374232533331381059609366957381520422
119659108818944863747230924803539970506096648792459211571201474029926645760175719883360665615273665358336009753326307394564541188003862401503501433711203359865448803645502536433052824295364817706105011928862
121159093058273281123428656480256502533711514828813736924509372416256015724101632003435265534309171234143187932463104254502903913758721422741928128998458912709239296001756335144196512356990934111984443414372275172541865828329161828205250506102
131390911669927936145243276302090247041490329010176021007112814880358443085118950631014464264601580856934471889215327738265661269217608867840039978164489786265619874202461876505674097460045724544200909075071124163745015653738060429575324987601228767924549613398392310260628142
141669094003913523218681013659185971297569778815310233631545876673965588487060854427414953984115894128952596234241441051998160571596813822335492560584704103116093235724574725982939099505944576267784774211758003290796803049686988822575835540886846438860044783493752414069943434794120589113209464971366238515473190982

Computed in your browser ?

Technical details
Node id
n-b5da320ca3ea5fd1 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-5eacb6f0303c8d92

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:          13,            9;
  1:         156,          225,            81;
  2:        1872,         4104,          2997,           729;
  3:       22464,        66096,         72900,         35721,          6561;
  4:      269568,       995328,       1469664,       1084752,        400221,         59049;
  5:     3234816,     14370048,      26593920,      26244000,      14565420,       4310577,        531441;
  6:    38817792,    201553920,     448457472,     554273280,     410981040,     182815704,      45172485,       4782969;
  7:   465813504,   2768007168,    7195474944,   10687396608,    9920232000,    5892617808,    2187411156,     463947993,     43046721;
  8:  5589762048,  37408407552,  111257763840,  193008033792,  215229353472,  159993501696,   79282494144,   25254076320,   4692092589,   387420489;
  9: 67077144576, 499208749056, 1671768834048, 3317416280064, 4319824545792, 3856986201600, 2391331444992, 1016591363136, 283591797948, 46877879169, 3486784401;
%F Formula
The row polynomials satisfy P(n,x) = (12+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 13 + 9*x.
%t Mathematica
P[0, x_] := 13 + 9*x;
P[n_, x_] := P[n, x] = Expand[(12 + 9*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] = 13 + 9*x; for(n = 1, N, my(i = n + 1); v[i] = (12 + 9*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-3fedd67ff8cfe3e1

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.