Hadamard-factor parent of A155495 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 8 + 16\,x + 4\,x^{2}\)

\(P_{2}(x) = 32 + 96\,x + 72\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	8	16	4															28
2	32	96	72	8														208
3	128	512	640	256	16													1552
4	512	2560	4480	3200	800	32												11584
5	2048	12288	27648	28672	13440	2304	64											86464
6	8192	57344	157696	215040	150528	50176	6272	128										645376
7	32768	262144	851968	1441792	1351680	688128	172032	16384	256									4817152
8	131072	1179648	4423680	8945664	10543104	7299072	2838528	552960	41472	512								35955712
9	524288	5242880	22282240	52428800	74547200	65601536	35143680	10813440	1689600	102400	1024							268377088
10	2097152	23068672	109576192	294125568	490209280	524812288	360808448	154632192	38658048	4956160	247808	2048						2003193856
11	8388608	100663296	528482304	1593835520	3048210432	3850371072	3244294144	1799356416	632586240	131203072	14057472	589824	4096					14952042496
12	33554432	436207616	2508193792	8396996608	18129879040	26417823744	26417823744	18075353088	8284536832	2436628480	426409984	38764544	1384448	8192				111603564544
13	134217728	1879048192	11744051200	43218108416	103993573376	171815469056	199149748224	162571223040	92462383104	35687235584	8921808896	1335885824	104366080	3211264	16384			833020346368
14	536870912	8053063680	54358179840	218103808000	578813952000	1069648183296	1411341352960	1341212590080	914463129600	440297062400	145298030592	31284264960	4055367680	275251200	7372800	32768		6217748512768
15	2147483648	34359738368	249108103168	1082331758592	3140694835200	6420976107520	9507983851520	10322953895936	8226103885824	4768755875840	1972530839552	563580239872	105671294976	11978932224	713031680	16777216	65536	46409906716672

Technical details

Node id: n-1e01398e35510f54 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-d44a70ca53ce7ac3

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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:       8,       16,         4;
   2:      32,       96,        72,         8;
   3:     128,      512,       640,       256,        16;
   4:     512,     2560,      4480,      3200,       800,        32;
   5:    2048,    12288,     27648,     28672,     13440,      2304,        64;
   6:    8192,    57344,    157696,    215040,    150528,     50176,      6272,       128;
   7:   32768,   262144,    851968,   1441792,   1351680,    688128,    172032,     16384,      256;
   8:  131072,  1179648,   4423680,   8945664,  10543104,   7299072,   2838528,    552960,    41472,     512;
   9:  524288,  5242880,  22282240,  52428800,  74547200,  65601536,  35143680,  10813440,  1689600,  102400,   1024;
  10: 2097152, 23068672, 109576192, 294125568, 490209280, 524812288, 360808448, 154632192, 38658048, 4956160, 247808, 2048;

%F Formula

The row polynomials satisfy P(n,x) = -4*x^2*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 8 + 16*x + 4*x^2, P(2,x) = 32 + 96*x + 72*x^2 + 8*x^3.

%t Mathematica

P[1, x_] := 8 + 16*x + 4*x^2;
P[2, x_] := 32 + 96*x + 72*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-4*x^2*P[n - 2, x] + (4 + 4*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] = 8 + 16*x + 4*x^2; v[2] = 32 + 96*x + 72*x^2 + 8*x^3; for(n = 3, N, my(i = n); v[i] = -4*x^2*v[i - 2] + (4 + 4*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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

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