Real-rooted atlas

A038281 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(8 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 624\,x + 225\,x^{2} + 27\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17Pn(1)
0576624225271452
14608672036728918115972
2368646758449536181443321243175692
329491265126459904029376081000119077291932612
42359296609484867461124147200152928033825641553218721258732
51887436855836672722534405341593624675840729388813471921421556561233846052
6150994944503316480745537536644087808357654528132378624326592005178816478953196832572306572
71207959552447951667274742497287389315072479349964821319925766584094721394081281936807215943235904928295372292
896636764163946001203273232547840815372697606051594240031436439552116632535043090493440573168960708588005255361177147311249095212
97730941132834467112550470424041881687199580160072873934848043303934361618761534668859713708032138568320002286377280254619288171832595314413423740047332
10618475290624298929723801666679367270409088687669248844590219264056505327943682800040804352104055570432028999578009659861514240889608614490132393655801305159432337661140520652
11494780232499225769803776000623113855303689271331153510494833280548864705419689328643935192481792016724568047616544163335372813488794542082507532318723389884992031503822481801584994782969414272545727172
1239582418599936221001837182976575800495570944928640648871936103680617899622484883559310950452644130534195225185231883468893706770972672271159356948486052664217600102345049497612689960774410892414736578739249143489074556998002998892
133166593487994881886761953263616526940947611648091565266776883201108037137858560099011032818647046758037222064128359414246670336015052111242854404980477984768001297691208253442634559661260840855483468804678381411203730715820018510090034304672150126978032987812
142533274790395904160440736725073924781556166872268889060441849856000116112551061749760112449940390674432837676076221071364902725139981926422824116394393600850001576067072025322963620331526000721353768961117211766128641599935016960017019716889601267295466245897400777129140163551396758362865932

Computed in your browser ?

Technical details
Node id
n-15f7f9eaa70b650d · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-0bc72b01ed588d8f

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,          624,          225,           27;
  1:        4608,         6720,         3672,          891,           81;
  2:       36864,        67584,        49536,        18144,         3321,          243;
  3:      294912,       651264,       599040,       293760,        81000,        11907,          729;
  4:     2359296,      6094848,      6746112,      4147200,      1529280,       338256,        41553,        2187;
  5:    18874368,     55836672,     72253440,     53415936,     24675840,      7293888,      1347192,      142155,        6561;
  6:   150994944,    503316480,    745537536,    644087808,    357654528,    132378624,     32659200,     5178816,      478953,      19683;
  7:  1207959552,   4479516672,   7474249728,   7389315072,   4793499648,   2131992576,    658409472,   139408128,    19368072,    1594323,     59049;
  8:  9663676416,  39460012032,  73232547840,  81537269760,  60515942400,  31436439552,  11663253504,  3090493440,   573168960,   70858800,   5255361,   177147;
  9: 77309411328, 344671125504, 704240418816, 871995801600, 728739348480, 433039343616, 187615346688, 59713708032, 13856832000, 2286377280, 254619288, 17183259, 531441;
%F Formula
The row polynomials satisfy P(n,x) = (8+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 624*x + 225*x^2 + 27*x^3.
%t Mathematica
P[0, x_] := 576 + 624*x + 225*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[(8 + 3*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 + 624*x + 225*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (8 + 3*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-b64329fdd6480cca

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.