Real-rooted atlas

A122076: Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.

A122076 nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence from OEIS A122076
\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1325
278217
3183015265
44710480242257
51233403551703521025
6322106814109323154824097
7843326252084396207953263216385
822079760182801878411440414484080265537
95778287466178574838558092622676021260992262145
1015127836202028402829802498151444885469013080181512021048577
11396032408126507051027466104813572041233718310593021384253014324194305
1210368268776820485323611528418072233304081858892725208193347335283432168216777217
1327144319506506349200123619601601145014508520941135043927001460030336050507654550195267108865
147106475499704194211504138792059305960602450804455066024227400966283527827805605607462059152242268435457

Generated b-files: A052539 (row sums), A122076 ?

Computed in your browser ?

Technical details
Node id
n-9b40f045d129e237 · revision 1 · JSON
Definition source
oeis_catalog:A122076
Fingerprint
f6-f83cf33d2c000fbc

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:    2;
  1:    3,     2;
  2:    7,     8,     2;
  3:   18,    30,    15,     2;
  4:   47,   104,    80,    24,     2;
  5:  123,   340,   355,   170,    35,     2;
  6:  322,  1068,  1410,   932,   315,    48,    2;
  7:  843,  3262,  5208,  4396,  2079,   532,   63,    2;
  8: 2207,  9760, 18280, 18784, 11440,  4144,  840,   80,  2;
  9: 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260, 99, 2;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*v[i - 2]); 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.

A122076: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    2;
  1:    3,     2;
  2:    7,     8,     2;
  3:   18,    30,    15,     2;
  4:   47,   104,    80,    24,     2;
  5:  123,   340,   355,   170,    35,     2;
  6:  322,  1068,  1410,   932,   315,    48,    2;
  7:  843,  3262,  5208,  4396,  2079,   532,   63,    2;
  8: 2207,  9760, 18280, 18784, 11440,  4144,  840,   80,  2;
  9: 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260, 99, 2;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-644899f192503d96

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

Used in 3 proof steps

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023