A125662 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(8 + 6\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x) + \left(4 + 8\,x + 3\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	3							3
2	24	18						42
3	138	216	81					435
4	696	1674	1296	324				3990
5	3273	10656	12636	6480	1215			34260
6	14736	60588	97200	76140	29160	4374		282198
7	64404	320112	648810	686880	400950	122472	15309	2258937

All 15 rows

Technical details

Node id: n-59b964c94fda9db6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-bd74b728e6b7daff

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:       3;
  2:      24,      18;
  3:     138,     216,       81;
  4:     696,    1674,     1296,      324;
  5:    3273,   10656,    12636,     6480,     1215;
  6:   14736,   60588,    97200,    76140,    29160,     4374;
  7:   64404,  320112,   648810,   686880,   400950,   122472,   15309;
  8:  275472, 1606284,  3938544,  5266620,  4150440,  1928934,  489888,   52488;
  9: 1159119, 7757856, 22333644, 36133776, 35950635, 22534848, 8695512, 1889568, 177147;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (4*n+8+(3*n+6)*x)*P(n-1,x) + (4+8*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (3*n + 6)*x)*P[n - 1, x] + (4 + 8*x + 3*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (3*n + 6)*x)*v[i - 1] + (4 + 8*x + 3*x^2)*deriv(v[i - 1], x))/(2*n - 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 (−∞, 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    
    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 reversal of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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