A085478 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(20 - 18\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 46\,x - 14\,n - 50\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x + 12\,x^{2} - 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	3	2													5
4	5	12	3												20
5	7	30	30	4											71
6	9	56	105	60	5										235
7	11	90	252	280	105	6									744
8	13	132	495	840	630	168	7								2285
9	15	182	858	1980	2310	1260	252	8							6865
10	17	240	1365	4004	6435	5544	2310	360	9						20284
11	19	306	2040	7280	15015	18018	12012	3960	495	10					59155
12	21	380	2907	12240	30940	48048	45045	24024	6435	660	11				170711
13	23	462	3990	19380	58140	111384	136136	102960	45045	10010	858	12			488400
14	25	552	5313	29260	101745	232560	352716	350064	218790	80080	15015	1092	13		1387225
15	27	650	6900	42504	168245	447678	813960	1007760	831402	437580	136136	21840	1365	14	3916061

Technical details

Node id: n-cd6dc20f7cfa6d5c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5848b624a80b5ea0

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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:  0;
   2:  1;
   3:  3,   2;
   4:  5,  12,    3;
   5:  7,  30,   30,    4;
   6:  9,  56,  105,   60,    5;
   7: 11,  90,  252,  280,  105,    6;
   8: 13, 132,  495,  840,  630,  168,    7;
   9: 15, 182,  858, 1980, 2310, 1260,  252,   8;
  10: 17, 240, 1365, 4004, 6435, 5544, 2310, 360, 9;

%F Formula

The row polynomials satisfy (4*n^2-18*n+20)*P(n,x) = (4*n^2-14*n+12+(12*n^2-50*n+46)*x)*P(n-1,x) - ((2*n-3)*x+(8*n-12)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 14*n + 12 + (12*n^2 - 50*n + 46)*x)*P[n - 1, x] - ((2*n - 3)*x + (8*n - 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 18*n + 20)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 14*n + 12 + (12*n^2 - 50*n + 46)*x)*v[i - 1] - ((2*n - 3)*x + (8*n - 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 18*n + 20)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n + k, 2*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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n + k, 2*k).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Three-term recurrence criterion (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023