chebyshev_t → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0)coefficients of one sign Hadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-1 - 2\,x + 2\,x^{2} + 6\,x^{3} + 3\,x^{4} + n + 2\,n x - 2\,n x^{2} - 6\,n x^{3} - 3\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} + 15\,x^{3} + 11\,x^{4} + 3\,x^{5}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	6	7													14
3	1	12	33	26												72
4	1	20	94	164	97											376
5	1	30	210	600	765	362										1968
6	1	42	405	1668	3411	3426	1351									10304
7	1	56	707	3906	11319	18004	14917	5042								53952
8	1	72	1148	8120	31110	69368	90236	63624	18817							282496
9	1	90	1764	15456	74862	218628	395700	435312	267129	70226						1479168
10	1	110	2595	27480	163050	596996	1405830	2139960	2039205	1107710	262087					7745024
11	1	132	3685	46266	328482	1462384	4291826	8457900	11104797	9332444	4547433	978122				40553472
12	1	156	5082	74492	621423	3288120	11670316	28534488	48316239	55757996	41907930	18514092	3650401			212340736
13	1	182	6838	115544	1116115	6896318	28948036	85246304	178458007	264781010	272530726	185254472	74853493	13623482		1111830528
14	1	210	9009	173628	1918917	13651638	66613365	231121512	580845699	1062469086	1402445331	1302543900	808138359	300846042	50843527	5821620224

Technical details

Node id: n-e9f3812507785414 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-73a02a3902b59fb9

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: 1,  2;
  2: 1,  6,    7;
  3: 1, 12,   33,    26;
  4: 1, 20,   94,   164,    97;
  5: 1, 30,  210,   600,   765,    362;
  6: 1, 42,  405,  1668,  3411,   3426,   1351;
  7: 1, 56,  707,  3906, 11319,  18004,  14917,   5042;
  8: 1, 72, 1148,  8120, 31110,  69368,  90236,  63624,  18817;
  9: 1, 90, 1764, 15456, 74862, 218628, 395700, 435312, 267129, 70226;

%F Formula

The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n-1+(2*n-2)*x-(2*n-2)*x^2-(6*n-6)*x^3-(3*n-3)*x^4)*P(n-1,x) + (2*x+9*x^2+15*x^3+11*x^4+3*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n - 2)*x - (2*n - 2)*x^2 - (6*n - 6)*x^3 - (3*n - 3)*x^4)*P[n - 1, x] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (2*n - 2)*x - (2*n - 2)*x^2 - (6*n - 6)*x^3 - (3*n - 3)*x^4)*v[i - 1] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^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 [-1, 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 shift_x_plus_one of chebyshev_t?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_t?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A086645 → basis A013609 → x ↦ x+1

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023