Real-rooted atlas

A038301 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(9 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 810 + 2871\,x + 3388\,x^{2} + 1331\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
08102871338813318400
1729034749620734924714641168000
26561039293194089611260266734861610513360000
359049042580891279030520484090184476608857805177156167200000
4531441044818191161951724325050165391353930282644505113379904194871711344000000
54782969046182222919505656174706920449709773718568486937754129508691142256348321435888126880000000
64304672104682526651226351350726381850582811565575960413971335301011250120974458227666948175774282422357947691537600000000
738742048904687787916925522400880982335303824417429054005442529633532734254935777080617615623097167987011906062145732398812593742460110752000000000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bd6fba845345add6 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-535287d0474f2922

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:          810,          2871,           3388,            1331;
  1:         7290,         34749,          62073,           49247,           14641;
  2:        65610,        392931,         940896,         1126026,          673486,          161051;
  3:       590490,       4258089,       12790305,        20484090,        18447660,         8857805,         1771561;
  4:      5314410,      44818191,      161951724,       325050165,       391353930,       282644505,       113379904,        19487171;
  5:     47829690,     461822229,     1950565617,      4706920449,      7097737185,      6848693775,      4129508691,      1422563483,       214358881;
  6:    430467210,    4682526651,    22635135072,     63818505828,    115655759604,    139713353010,    112501209744,     58227666948,     17577428242,      2357947691;
  7:   3874204890,   46877879169,   255224008809,    823353038244,   1742905400544,   2529633532734,   2549357770806,   1761562309716,    798701190606,    214573239881,     25937424601;
  8:  34867844010,  464517166311,  2812672750140,  10217641441095,  24743032025580,  41938661200590,  50770188797328,  43896996266310,  26565496122330,  10716872255595,   2593742460100,   285311670611;
  9: 313810596090, 4564200780909, 30423743580681, 122898173221395, 335081344082265, 649621303086690, 918256972382442, 953545043167398, 721956424030380, 388672307645985, 141229276952445, 31098972096599, 3138428376721;
%F Formula
The row polynomials satisfy P(n,x) = (9+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 810 + 2871*x + 3388*x^2 + 1331*x^3.
%t Mathematica
P[0, x_] := 810 + 2871*x + 3388*x^2 + 1331*x^3;
P[n_, x_] := P[n, x] = Expand[(9 + 11*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] = 810 + 2871*x + 3388*x^2 + 1331*x^3; for(n = 1, N, my(i = n + 1); v[i] = (9 + 11*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-0ccb5845893f0f92

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.