A084534 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 + 16\,x + 32\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 16\,x + 3\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 8\,x\)

\(P_{2}(x) = 7 + 32\,x + 32\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	8														11
2	7	32	32													71
3	18	120	240	128												506
4	47	416	1280	1536	512											3791
5	123	1360	5680	10880	8960	2048										29051
6	322	4272	22560	59648	80640	49152	8192									224786
7	843	13048	83328	281344	532224	544768	258048	32768								1746371
8	2207	39040	292480	1202176	2928640	4243456	3440640	1310720	131072							13590431
9	5778	114984	988560	4789632	14287104	26855424	31137792	20643840	6488064	524288						105835466
10	15127	334480	3245440	18110720	63952640	147955712	224010240	214302720	118947840	31457280	2097152					824429351
11	39603	963248	10411280	65757824	268322560	737701888	1381101568	1735557120	1401421824	663224320	149946368	8388608				6422836211
12	103682	2751072	32776512	231137792	1070264832	3410337792	7614021632	11881807872	12671188992	8789164032	3598712832	704643072	33554432			50040464546
13	271443	7802600	101587200	791165440	4098931200	14856724480	38548889600	71969996800	95684526080	88093491200	53230960640	19084083200	3271557120	134217728		389874204731
14	710647	21998816	310738400	2648826880	15182325760	61690961920	182479503360	396941721600	633263554560	729489080320	587789762560	312978964480	99237232640	15032385536	536870912	3037604638391

Technical details

Node id: n-8018092894e25502 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-261d320b22506b21

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:    3,      8;
  2:    7,     32,     32;
  3:   18,    120,    240,     128;
  4:   47,    416,   1280,    1536,      512;
  5:  123,   1360,   5680,   10880,     8960,     2048;
  6:  322,   4272,  22560,   59648,    80640,    49152,     8192;
  7:  843,  13048,  83328,  281344,   532224,   544768,   258048,    32768;
  8: 2207,  39040, 292480, 1202176,  2928640,  4243456,  3440640,  1310720,  131072;
  9: 5778, 114984, 988560, 4789632, 14287104, 26855424, 31137792, 20643840, 6488064, 524288;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2+(8*n-16)*x+(16*n-32)*x^2)*P(n-2,x) + (3*n-6+(8*n-16)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 7 + 32*x + 32*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[2, x_] := 7 + 32*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*P[n - 2, x] + (3*n - 6 + (8*n - 16)*x)*P[n - 1, x])/(n - 2)]];
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] = 3 + 8*x; v[3] = 7 + 32*x + 32*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*v[i - 2] + (3*n - 6 + (8*n - 16)*x)*v[i - 1])/(n - 2)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A086645 → basis A013611 → basis A013611

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