A091042 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 36\,x - 36\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 12\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 13 + 18\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	13	18														31
3	121	300	180													601
4	1093	3822	4284	1512												10711
5	9841	44280	71928	49248	11664											186961
6	88573	487146	1035144	1050192	498960	85536										3245551
7	797161	5181540	13601484	18296928	13124592	4650048	606528									56258281
8	7174453	53808390	168151140	281932920	271246320	147456288	40824000	4199040								974792551
9	64570081	548845680	1989565488	3996281664	4829194080	3556493568	1532369664	342641664	28553472							16888515361
10	581130733	5520741954	22773060432	53309663136	77603077728	72273705408	42560909568	14998970880	2776405248	191476224						292589141311
11	5230176601	54916854300	253990450980	679024019520	1156793490720	1304248417920	977217478272	474437772288	139994313984	21868600320	1269789696					5068991364601
12	47071589413	541323278238	2774281800780	8339761752840	16278817393440	21560684048064	19644543418752	12201950585088	4996053063936	1256980944384	168277367808	8344332288				87818089575031
13	423644304721	5295553809000	29787490175400	99457119972000	218865652614000	333124983072000	357781176748800	271351314662400	142864399814400	50206073702400	10928051988480	1269789696000	54419558400			1521409670118001
14	3812798742493	51472783023642	315270796019544	1157601443204592	2835527041332720	4877322656389344	6036288621689088	5413681523538432	3493421512379136	1585992574932480	485129683703808	92450122592256	9423290732544	352638738432		26357747487018511
15	34315188682441	497570235895380	3296402812806588	13201160321094048	35623890892949904	68341806399142080	95799064618127808	99257495110142976	75963844117119744	42417363513388032	16834318221726720	4533800798232576	764022301765632	68934343016448	2272560758784	456636261434849161

Technical details

Node id: n-2b8ae16f62b5e385 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-78665ffd6c0972be

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:        13,         18;
   3:       121,        300,         180;
   4:      1093,       3822,        4284,        1512;
   5:      9841,      44280,       71928,       49248,       11664;
   6:     88573,     487146,     1035144,     1050192,      498960,       85536;
   7:    797161,    5181540,    13601484,    18296928,    13124592,     4650048,      606528;
   8:   7174453,   53808390,   168151140,   281932920,   271246320,   147456288,    40824000,     4199040;
   9:  64570081,  548845680,  1989565488,  3996281664,  4829194080,  3556493568,  1532369664,   342641664,   28553472;
  10: 581130733, 5520741954, 22773060432, 53309663136, 77603077728, 72273705408, 42560909568, 14998970880, 2776405248, 191476224;

%F Formula

The row polynomials satisfy P(n,x) = -(9+36*x+36*x^2)*P(n-2,x) + (10+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 13 + 18*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 13 + 18*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 36*x + 36*x^2)*P[n - 2, x] + (10 + 12*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] = 13 + 18*x; for(n = 3, N, my(i = n); v[i] = -(9 + 36*x + 36*x^2)*v[i - 2] + (10 + 12*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 of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    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 reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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