A327997 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 4\,x\)

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	4														5
2	1	9	17													27
3	1	15	65	78												159
4	1	22	161	452	393											1029
5	1	30	325	1560	3229	2208										7353
6	1	39	580	4155	14854	24489	13977									58095
7	1	49	952	9415	50554	144949	200745	100026								506691
8	1	60	1470	19068	141939	613788	1483250	1793952	806769							4860297
9	1	72	2166	35532	347907	2091096	7635410	16114176	17516313	7280604						51023277
10	1	85	3075	62070	770133	6099051	30987305	98802230	186952851	186555969	72865089					583097859
11	1	99	4235	102960	1574133	15814953	105936809	469891620	1341121771	2320470603	2159159265	801693126				7215769575
12	1	114	5687	163680	3017058	37378638	317746715	1857675072	7366915006	19176823578	30796774227	27032735124	9620848953			96210083853
13	1	130	7475	251108	5483478	81945006	859136135	6372284204	33275991034	120131365354	289345567719	436244181048	364414877397	125072630712		1375803720801
14	1	147	9646	373737	9530521	168780612	2134535975	19519831617	129212700617	613373885124	2044269093503	4607222712729	6579491760360	5265848876625	1751021612937	21012273704151

Technical details

Node id: n-865eaaf10b6f92c2 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-8e85c3e2c9748635

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,  4;
  2: 1,  9,   17;
  3: 1, 15,   65,    78;
  4: 1, 22,  161,   452,    393;
  5: 1, 30,  325,  1560,   3229,    2208;
  6: 1, 39,  580,  4155,  14854,   24489,   13977;
  7: 1, 49,  952,  9415,  50554,  144949,  200745,   100026;
  8: 1, 60, 1470, 19068, 141939,  613788, 1483250,  1793952,   806769;
  9: 1, 72, 2166, 35532, 347907, 2091096, 7635410, 16114176, 17516313, 7280604;

%F Formula

The row polynomials satisfy P(n,x) = -(3*n-3)*x^2*P(n-2,x) + (1+(n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(3*n - 3)*x^2*P[n - 2, x] + (1 + (n + 3)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n - 3)*x^2*v[i - 2] + (1 + (n + 3)*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 basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Geometric basis map (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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - 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 (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A327997 → x ↦ x+1

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