A034807: Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.

A034807 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)basis map preserves real roots

Recurrence from A034807

\(P_{n}(x) = \left(1\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	1	2							3
3	1	3							4
4	1	4	2						7
5	1	5	5						11
6	1	6	9	2					18
7	1	7	14	7					29
8	1	8	20	16	2				47
9	1	9	27	30	9				76
10	1	10	35	50	25	2			123
11	1	11	44	77	55	11			199
12	1	12	54	112	105	36	2		322
13	1	13	65	156	182	91	13		521
14	1	14	77	210	294	196	49	2	843

b-files: A000032 (row sums), A000204 (row sums), A034807 ?

Technical details

Node id: n-fe6a74fcc4c3def7 · revision 0 · JSON
Definition source: oeis_catalog:A034807
Fingerprint: f6-088313a0be276231

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

Matches ?

A034807 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A254729, A329723, A000204, A080023, A000032, A222333, A358995, A169985

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2;
  1: 1;
  2: 1, 2;
  3: 1, 3;
  4: 1, 4,  2;
  5: 1, 5,  5;
  6: 1, 6,  9,  2;
  7: 1, 7, 14,  7;
  8: 1, 8, 20, 16, 2;
  9: 1, 9, 27, 30, 9;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + x*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)

polytool c48629374597

A034807 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 2;
  1: 1;
  2: 1, 2;
  3: 1, 3;
  4: 1, 4,  2;
  5: 1, 5,  5;
  6: 1, 6,  9,  2;
  7: 1, 7, 14,  7;
  8: 1, 8, 20, 16, 2;
  9: 1, 9, 27, 30, 9;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + x*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + x*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a real-rooted image with every root <= -c/b = 0
by Constant-current basis map
Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a PF image (zero allowed)
by Constant-current basis map

Proof chains ? JSON

Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? step › direct check
Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 10

2 more

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	1	2							3
3	1	3							4
4	1	4	2						7
5	1	5	5						11
6	1	6	9	2					18
7	1	7	14	7					29
8	1	8	20	16	2				47
9	1	9	27	30	9				76
10	1	10	35	50	25	2			123
11	1	11	44	77	55	11			199
12	1	12	54	112	105	36	2		322
13	1	13	65	156	182	91	13		521
14	1	14	77	210	294	196	49	2	843

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	1	2							3
3	1	3							4
4	1	4	2						7
5	1	5	5						11
6	1	6	9	2					18
7	1	7	14	7					29
8	1	8	20	16	2				47
9	1	9	27	30	9				76
10	1	10	35	50	25	2			123
11	1	11	44	77	55	11			199
12	1	12	54	112	105	36	2		322
13	1	13	65	156	182	91	13		521
14	1	14	77	210	294	196	49	2	843

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	1								1
2	1	2							3
3	1	3							4
4	1	4	2						7
5	1	5	5						11
6	1	6	9	2					18
7	1	7	14	7					29
8	1	8	20	16	2				47
9	1	9	27	30	9				76
10	1	10	35	50	25	2			123
11	1	11	44	77	55	11			199
12	1	12	54	112	105	36	2		322
13	1	13	65	156	182	91	13		521
14	1	14	77	210	294	196	49	2	843