A207824 → basis A013609 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1/2, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-3\,x - 8\,x^{2} + 2\,n + 9\,n x + 8\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 8\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	6							8
2	4	24	35						63
3	8	72	212	204					496
4	16	192	852	1656	1189				3905
5	32	480	2848	8352	12102	6930			30744
6	64	1152	8560	33600	73464	84816	40391		242047
7	128	2688	24000	118080	345680	602064	577576	235416	1905632

All 15 rows

Technical details

Node id: n-d0cb59aeab637630 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-611b245e25712c49

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,     6;
  2:   4,    24,     35;
  3:   8,    72,    212,     204;
  4:  16,   192,    852,    1656,    1189;
  5:  32,   480,   2848,    8352,   12102,     6930;
  6:  64,  1152,   8560,   33600,   73464,    84816,    40391;
  7: 128,  2688,  24000,  118080,  345680,   602064,   577576,   235416;
  8: 256,  6144,  64064,  379008, 1391280,  3244608,  4694072,  3851472,  1372105;
  9: 512, 13824, 164864, 1139712, 5032608, 14718816, 28510112, 35265312, 25275466, 7997214;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n+(9*n-3)*x+(8*n-8)*x^2)*P(n-1,x) - (x+6*x^2+8*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 + (9*n - 3)*x + (8*n - 8)*x^2)*P[n - 1, x] - (x + 6*x^2 + 8*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 + (9*n - 3)*x + (8*n - 8)*x^2)*v[i - 1] - (x + 6*x^2 + 8*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 (-1/2, 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 Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).?
  - 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 ?

A207824 → basis A013609

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