Real-rooted atlas

A038277 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(7 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8 + 11\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
081119
156165121342
23921771266213316156
3274416709381153859914641110808
4192081471474506046894585270761610511994544
5134456124131747728459782850112735706925193177156135901792
694119210168235470644021209812451865263401724856218857805019487171646232256
76588344815307574413013991364577137263647807532591890872517388181111076874721435888111632180608
84611840864318708339859481201440635534833465695032518155824345347279722435466651220137189683842357947691209379250944
9322828856500961206934976694753144689916756392729774052730831722390944280987342836467328004486165942108167414286061259374246013768826516992
102259801992386184018992999425960301397573059575434069750268094358495713021464911585768416242362156790126043022028006519725365615202311911887828531167061167838877305856
111581861394429518663520525244005930991308237997335545758186174085113798619528594206338156288110274836809532054266896099144290184285401790105858788128539112425149200193531384283767211221099791505408
1211073029760822403111998192091785713894811934506633757446421348292550012999303846150932696151908831304419357738589358848914775988626244225854903117925262829110966853211144273854065662887354106583323452271214393121979796247097344
13775112083256169002116724211710684231706451065511892891446456229011019181414205861004486151331722975925851515901271269839946080369694436867836833872379093143396488244170197689936712193904199814142798491140805503417748502249169379749833583241395636332447752192
14542578458279212682771462276313833812905911469340335905117217436566615931486041496262182435130113884705541974437187779842624072328811211727700740468912146777730417074656913714367089162960229706922160511445875037930767447618041810025797706302334025348235982354641772481694156517121453984059539456

Computed in your browser ?

Technical details
Node id
n-d3ee9b6e124c7b6d · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-4fd62f6b79b2875b

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:         8,         11;
  1:        56,        165,         121;
  2:       392,       1771,        2662,         1331;
  3:      2744,      16709,       38115,        38599,        14641;
  4:     19208,     147147,      450604,       689458,       527076,       161051;
  5:    134456,    1241317,     4772845,      9782850,     11273570,      6925193,      1771561;
  6:    941192,   10168235,    47064402,    120981245,    186526340,    172485621,     88578050,     19487171;
  7:   6588344,   81530757,   441301399,   1364577137,   2636478075,   3259189087,   2517388181,   1110768747,    214358881;
  8:  46118408,  643187083,  3985948120,  14406355348,  33465695032,  51815582434,  53472797224,  35466651220,  13718968384,   2357947691;
  9: 322828856, 5009612069, 34976694753, 144689916756, 392729774052, 730831722390, 944280987342, 836467328004, 486165942108, 167414286061, 25937424601;
%F Formula
The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 11*x.
%t Mathematica
P[0, x_] := 8 + 11*x;
P[n_, x_] := P[n, x] = Expand[(7 + 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] = 8 + 11*x; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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-3667cd139473d050

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.