A327997 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(3\,x^{2} - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(1 + 2\,x + n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	1	7	9													17
3	1	12	38	27												78
4	1	18	101	192	81											393
5	1	25	215	755	969	243										2208
6	1	33	400	2205	5494	5115	729									13977
7	1	42	679	5355	21469	40971	29322	2187								100026
8	1	52	1078	11452	66619	209356	323658	187992	6561							806769
9	1	63	1626	22302	176295	813645	2111318	2764926	1370745	19683						7280604
10	1	75	2355	40410	414813	2619981	10076345	22448130	25811091	11392839	59049					72865089
11	1	88	3300	69135	891363	7343490	38847248	129031265	254297241	263989242	107043606	177147				801693126
12	1	102	4499	112860	1781538	18489042	127967279	586433364	1728215566	3083362326	2951126991	1122823944	531441			9620848953
13	1	117	5993	177177	3355638	42723252	373213841	2241576909	9126215098	24361470276	40046861205	35886116097	12989320785	1594323		125072630712
14	1	133	7826	269087	6015009	92011920	987305891	7491925727	40000721761	147510010648	362429978547	556384764663	472073225688	164040593067	4782969	1751021612937

Technical details

Node id: n-9aa91d153ab99ae4 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-77231177ad534f16

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  7,    9;
  3: 1, 12,   38,    27;
  4: 1, 18,  101,   192,     81;
  5: 1, 25,  215,   755,    969,    243;
  6: 1, 33,  400,  2205,   5494,   5115,     729;
  7: 1, 42,  679,  5355,  21469,  40971,   29322,    2187;
  8: 1, 52, 1078, 11452,  66619, 209356,  323658,  187992,    6561;
  9: 1, 63, 1626, 22302, 176295, 813645, 2111318, 2764926, 1370745, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(3*n-3)*x^2*P(n-2,x) + (1+(n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(3*n - 3)*x^2*P[n - 2, x] + (1 + (n + 2)*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] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n - 3)*x^2*v[i - 2] + (1 + (n + 2)*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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

Related ?

Same fingerprint ?

A327997: Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

References

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