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 ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	3	2							5
2	7	8	2						17
3	18	30	15	2					65
4	47	104	80	24	2				257
5	123	340	355	170	35	2			1025
6	322	1068	1410	932	315	48	2		4097
7	843	3262	5208	4396	2079	532	63	2	16385

Show all 15 stored rows

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 ?

A123166: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A122076: triangle, 53 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A046231: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A052539: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A092896: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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

theorem_check three_term_recurrence e-644899f192503d96 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 3 proof steps

e-58ea4b3a2d7c4054 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices)…
e-895e752df1147f88 preserver_image (affine, shift 0) → shift_x_plus_one of Coefficients of a generalized Jaco-Lucas polynomial (even indices) re…
e-fa6dba774ff7b5f5 preserver_image (derivative_polynomial, shift 0) → derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by …

Related sequences ?

No candidate equalities or related sequences.

References

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