A091042 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(8 + 14\,x - 20\,n - 34\,n x + 8\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 4\,x^{2} - 4\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	6							6
3	40	40						80
4	224	448	168					840
5	1152	3456	2880	576				8064
6	5632	22528	29568	14080	1760			73568
7	26624	133120	239616	186368	58240	4992		648960
8	122880	737280	1689600	1843200	967680	215040	13440	5589120

All 15 rows

Technical details

Node id: n-b3c12e32b1daabea · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d6d34c33242db792

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:       6;
   3:      40,       40;
   4:     224,      448,      168;
   5:    1152,     3456,     2880,       576;
   6:    5632,    22528,    29568,     14080,      1760;
   7:   26624,   133120,   239616,    186368,     58240,     4992;
   8:  122880,   737280,  1689600,   1843200,    967680,   215040,    13440;
   9:  557056,  3899392, 10862592,  15319040,  11489280,  4386816,   731136,   34816;
  10: 2490368, 19922944, 65372160, 113311744, 111288320, 61636608, 17977344, 2334720, 87552;

%F Formula

The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (8*n^2-20*n+8+(12*n^2-34*n+14)*x)*P(n-1,x) - ((4*n-2)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 20*n + 8 + (12*n^2 - 34*n + 14)*x)*P[n - 1, x] - ((4*n - 2)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 20*n + 8 + (12*n^2 - 34*n + 14)*x)*v[i - 1] - ((4*n - 2)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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 of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    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 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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