A085841 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-4 - 3\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	16	40	5													61
3	64	336	140	7												547
4	256	2304	2016	336	9											4921
5	1024	14080	21120	7392	660	11										44287
6	4096	79872	183040	109824	20592	1144	13									398581
7	16384	430080	1397760	1281280	411840	48048	1820	15								3587227
8	65536	2228224	9748480	12673024	6223360	1244672	99008	2720	17							32285041
9	262144	11206656	63504384	111132672	77395968	23648768	3224832	186048	3876	19						290565367
10	1048576	55050240	392232960	889061376	833495040	361181184	75246080	7441920	325584	5320	21					2615088301
11	4194304	265289728	2321285120	6615662592	8033304576	4686094336	1384527872	209200640	15690048	538384	7084	23				23535794707
12	16777216	1258291200	13264486400	46425702400	70882099200	53555363840	21300428800	4564377600	523001600	30764800	850080	9200	25			211822152361
13	67108864	5888802816	73610035200	310388981760	581979340800	552880373760	284817162240	82158796800	13350804480	1199827200	56833920	1291680	11700	27		1906399371247
14	268435456	27246198784	398475657216	1992378286080	4500640235520	5250746941440	3401051996160	1270722723840	277970595840	35427624960	2563841280	99889920	1900080	14616	29	17157594341221

Technical details

Node id: n-9130684ebb51c694 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2c0f1fe74efc2e1a

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:      4,        3;
  2:     16,       40,        5;
  3:     64,      336,      140,         7;
  4:    256,     2304,     2016,       336,        9;
  5:   1024,    14080,    21120,      7392,      660,       11;
  6:   4096,    79872,   183040,    109824,    20592,     1144,      13;
  7:  16384,   430080,  1397760,   1281280,   411840,    48048,    1820,     15;
  8:  65536,  2228224,  9748480,  12673024,  6223360,  1244672,   99008,   2720,   17;
  9: 262144, 11206656, 63504384, 111132672, 77395968, 23648768, 3224832, 186048, 3876, 19;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = (8*n-4+(6*n-3)*x)*P(n-1,x) + (16*x-4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (6*n - 3)*x)*P[n - 1, x] + (16*x - 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n - 4 + (6*n - 3)*x)*v[i - 1] + (16*x - 4*x^2)*deriv(v[i - 1], x))/(2*n - 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 Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).?
  - 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 ?

A085841: Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+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