A086645 → basis A013610 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 72\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 24\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 5 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	12														17
3	41	168	144													353
4	365	2196	3888	1728												8177
5	3281	26256	72288	76032	20736											198593
6	29525	295260	1107360	1883520	1347840	248832										4912337
7	265721	3188664	15146352	35741952	42612480	22394880	2985984									122336033
8	2391485	33480804	192514896	581732928	976727808	876137472	355332096	35831808								3054149297
9	21523361	344373792	2324523456	8566304256	18590860800	23786348544	16805117952	5446434816	429981696							76315468673
10	193710245	3486784428	27022579776	117533700864	313102444032	522064963584	531994853376	305716985856	81266540544	5159780352						1907542343057
11	1743392201	34867844040	305093635920	1529826670080	4833010229760	9948941365248	13308560547840	11140825743360	5333922938880	1186749480960	61917364224					47685459212513
12	15690529805	345191655732	3365618644080	19114987953600	69874342433280	171314845599744	284396815245312	314581273509888	221368766496768	89960770437120	17027275161600	743008370688				1192108586037617
13	141214768241	3389154437808	36433410207264	231168575635200	959925039333120	2732320425074688	5423470150533120	7480204953255936	6997060750344192	4213455996321792	1475243120001024	240734712102912	8916100448256			29802463602463553
14	1270932914165	33044255768316	388270005278688	2722020568944768	12657498909269760	41036060663829504	94854246176342016	156988193219739648	183900214395076608	148016116464353280	77367851788271616	23626180171137024	3361369868992512	106993205379072		745059330625296977
15	11438396227481	320275094369496	4083507453212208	31346924861452032	161373608877793536	587146818911643648	1549640410071183360	2995118003801161728	4224190776160223232	4274907318864838656	3001842545162452992	1377968464110551040	370793869341622272	46435051134517248	1283918464548864	18626462930705797793

Technical details

Node id: n-3d116785edf34455 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5105b4356f23af0f

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         5,         12;
   3:        41,        168,         144;
   4:       365,       2196,        3888,         1728;
   5:      3281,      26256,       72288,        76032,        20736;
   6:     29525,     295260,     1107360,      1883520,      1347840,       248832;
   7:    265721,    3188664,    15146352,     35741952,     42612480,     22394880,      2985984;
   8:   2391485,   33480804,   192514896,    581732928,    976727808,    876137472,    355332096,     35831808;
   9:  21523361,  344373792,  2324523456,   8566304256,  18590860800,  23786348544,  16805117952,   5446434816,   429981696;
  10: 193710245, 3486784428, 27022579776, 117533700864, 313102444032, 522064963584, 531994853376, 305716985856, 81266540544, 5159780352;

%F Formula

The row polynomials satisfy P(n,x) = -(9+72*x+144*x^2)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 12*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 72*x + 144*x^2)*P[n - 2, x] + (10 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 12*x; for(n = 3, N, my(i = n); v[i] = -(9 + 72*x + 144*x^2)*v[i - 2] + (10 + 24*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ 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: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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