A082985 → reverse → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5	2						7
3	14	14	3					31
4	30	54	27	4				115
5	55	154	132	44	5			390
6	91	364	468	260	65	6		1254
7	140	756	1350	1100	450	90	7	3893

All 15 rows

Technical details

Node id: n-7a5a30d70173c719 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2bf8c8295f2d680d

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 -1)
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:   0;
  1:   1;
  2:   5,    2;
  3:  14,   14,    3;
  4:  30,   54,   27,     4;
  5:  55,  154,  132,    44,    5;
  6:  91,  364,  468,   260,   65,    6;
  7: 140,  756, 1350,  1100,  450,   90,    7;
  8: 204, 1428, 3366,  3740, 2210,  714,  119,   8;
  9: 285, 2508, 7524, 10868, 8645, 3990, 1064, 152, 9;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 3*n + 1 + (n^2 + n)*x)*v[i - 1] + (4*n*x + n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from shift_x_plus_one of Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A111125 → D

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