Real-rooted atlas

A038322 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(11 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 12 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
012820
113218464380
21452308021765127220
3159724549648576230404096137180
4175692628232898304642048229376327682606420
51932612831608814907200142489607659520219545626214449521980
62125873210693786423050790427599616019824640085426176204472322097152940917620
7233846052134638636033910898564880020992438867968025256591369083289601866465281677721617877434780
8257230657216681018376480730792968080894976087315644416628916879363019689164893197434881677721600134217728339671260820
928295372292204069654712662252019264127348308172816069436866561390333722624835299311616344092311552930128855041489816780810737418246453753955580
1031124909521224711291801688917329449600193063300531202786424520704028149220442112203109622087681046740992000037758802329609079829299201309965025288589934592122621325156020
113423740047332296724137435441178596573869442837082661811204609573377024005325553865195524486143478333442776292067901441252739619225604019485409280087048249671681142461300736687194767362329805177964380
12376611405206523537864715576401533835541204736406366818708787273401968441753609545767953334272919520091832320066428360573583363599047235469312144433509040128041731190738124882205674045440989560464998454975581388844266298381323220
13414272545727172419294031129925619702482725713216569710343876044801132515107826319361637250222400798721775133537282293761466328039775272969273220804902912044680063878168576161451117044039684242757673549824766497043513344852121511526404398046511104841059669245141180
144556998002998892494365237901091922502708324732394247843012400693550081701534893709787136270698733090193408032624470689311621123033067673578635264219311672035953868812333383670520872965350367397737922561758312280442798084237352886704537670693100107857927300757208432643518437208883215980133715657682420

Computed in your browser ?

Technical details
Node id
n-92d05df66a553ff7 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-f7279d99fdb813e1

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:          12,            8;
  1:         132,          184,           64;
  2:        1452,         3080,         2176,           512;
  3:       15972,        45496,        48576,         23040,          4096;
  4:      175692,       628232,       898304,        642048,        229376,         32768;
  5:     1932612,      8316088,     14907200,      14248960,       7659520,       2195456,       262144;
  6:    21258732,    106937864,    230507904,     275996160,     198246400,      85426176,     20447232,      2097152;
  7:   233846052,   1346386360,   3391089856,    4880020992,    4388679680,    2525659136,    908328960,    186646528,    16777216;
  8:  2572306572,  16681018376,  48073079296,   80808949760,   87315644416,   62891687936,  30196891648,   9319743488,  1677721600,   134217728;
  9: 28295372292, 204069654712, 662252019264, 1273483081728, 1606943686656, 1390333722624, 835299311616, 344092311552, 93012885504, 14898167808, 1073741824;
%F Formula
The row polynomials satisfy P(n,x) = (11+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 8*x.
%t Mathematica
P[0, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[(11 + 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] = 12 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = (11 + 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-991e7bd9709a3127

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.