A125662 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	4	16	15						35
3	8	48	92	56					204
4	16	128	372	464	209				1189
5	32	320	1248	2368	2182	780			6930
6	64	768	3760	9600	13464	9824	2911		40391
7	128	1792	10560	33920	64080	71136	42936	10864	235416

All 15 rows

Technical details

Node id: n-3373029d649348a1 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-c761722d499074dd

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:   2,    4;
  2:   4,   16,    15;
  3:   8,   48,    92,     56;
  4:  16,  128,   372,    464,    209;
  5:  32,  320,  1248,   2368,   2182,     780;
  6:  64,  768,  3760,   9600,  13464,    9824,    2911;
  7: 128, 1792, 10560,  33920,  64080,   71136,   42936,   10864;
  8: 256, 4096, 28224, 109312, 260080,  388992,  356952,  183648,  40545;
  9: 512, 9216, 72704, 329728, 946848, 1784384, 2205792, 1723968, 772746, 151316;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n+(6*n-2)*x+(3*n-3)*x^2)*P(n-1,x) - (x+4*x^2+3*x^3)*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[((2*n + (6*n - 2)*x + (3*n - 3)*x^2)*P[n - 1, x] - (x + 4*x^2 + 3*x^3)*D[P[n - 1, x], x])/(n)]];
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] = ((2*n + (6*n - 2)*x + (3*n - 3)*x^2)*v[i - 1] - (x + 4*x^2 + 3*x^3)*deriv(v[i - 1], x))/(n)); 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 basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from shift_x_plus_one of chebyshev_u?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_u?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

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