A111125 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(12 - 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-4\,n - 4\,n x + 6\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 12\,x - 4\,x^{2} + 5\,n + 12\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	2							2
3	14	8						22
4	62	80	24					166
5	230	480	312	64				1086
6	780	2280	2352	1024	160			6596
7	2508	9480	13584	9280	3040	384		38276
8	7786	36192	66840	63040	32160	8448	896	215362

All 15 rows

Technical details

Node id: n-de3bd3e4629dbe63 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-cfbb7597e7d40f64

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     0;
   2:     2;
   3:    14,      8;
   4:    62,     80,      24;
   5:   230,    480,     312,      64;
   6:   780,   2280,    2352,    1024,     160;
   7:  2508,   9480,   13584,    9280,    3040,    384;
   8:  7786,  36192,   66840,   63040,   32160,   8448,    896;
   9: 23578, 130240,  295512,  358080,  251040, 102144,  22400,  2048;
  10: 70090, 448840, 1210416, 1799232, 1616160, 900480, 304640, 57344, 4608;

%F Formula

The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (6*n^2-4*n+(4*n^2-4*n)*x)*P(n-1,x) + (5*n-5+(12*n-12)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 4*n + (4*n^2 - 4*n)*x)*P[n - 1, x] + (5*n - 5 + (12*n - 12)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((6*n^2 - 4*n + (4*n^2 - 4*n)*x)*v[i - 1] + (5*n - 5 + (12*n - 12)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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