A085478 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(36 - 6\,x^{2} - 60\,n + 7\,n x^{2} + 39\,n^{2} - 2\,n^{2} x^{2} - 13\,n^{3} + 2\,n^{4}\right)\,P_{n}(x) = \left(-12\,n + 4\,n x^{2} + n x^{3} + 11\,n^{2} + n^{2} x - 3\,n^{2} x^{2} - n^{2} x^{3} - 5\,n^{3} - 2\,n^{3} x + 2\,n^{4} + n^{4} x\right)\,P_{n-1}(x) + \left(-8\,x + 6\,x^{2} + 6\,x^{3} + x^{4} + 16\,n x - 4\,n x^{2} - 6\,n x^{3} - n x^{4} - 12\,n^{2} x - 3\,n^{2} x^{2} + 4\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	P_n(1)
0														0
1														0
2	1													1
3	5	2												7
4	16	16	3											35
5	40	72	33	4										149
6	85	238	192	56	5									576
7	161	644	792	400	85	6								2088
8	280	1512	2610	1980	720	120	7							7229
9	456	3192	7326	7700	4160	1176	161	8						24179
10	705	6204	18216	25168	18655	7770	1792	208	9					78727
11	1045	11286	41184	72072	69615	39480	13328	2592	261	10				250873
12	1496	19448	86229	185900	225680	165648	75684	21432	3600	320	11			785448
13	2080	32032	169455	440440	654160	599760	352716	134520	32760	4840	385	12		2423160
14	2821	50778	315744	972400	1730430	1930248	1410864	689472	225225	48070	6336	456	13	7382857

Technical details

Node id: n-fc921724cf3da1bf · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e85e327a9afcf748

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   0;
  2:   1;
  3:   5,    2;
  4:  16,   16,    3;
  5:  40,   72,   33,    4;
  6:  85,  238,  192,   56,    5;
  7: 161,  644,  792,  400,   85,    6;
  8: 280, 1512, 2610, 1980,  720,  120,   7;
  9: 456, 3192, 7326, 7700, 4160, 1176, 161, 8;

%F Formula

The row polynomials satisfy (2*n^4-13*n^3+39*n^2-60*n+36-(2*n^2-7*n+6)*x^2)*P(n,x) = (2*n^4-5*n^3+11*n^2-12*n+(n^4-2*n^3+n^2)*x-(3*n^2-4*n)*x^2-(n^2-n)*x^3)*P(n-1,x) + ((4*n^3-12*n^2+16*n-8)*x+(n^3-3*n^2-4*n+6)*x^2-(6*n-6)*x^3-(n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^4 - 5*n^3 + 11*n^2 - 12*n + (n^4 - 2*n^3 + n^2)*x - (3*n^2 - 4*n)*x^2 - (n^2 - n)*x^3)*P[n - 1, x] + ((4*n^3 - 12*n^2 + 16*n - 8)*x + (n^3 - 3*n^2 - 4*n + 6)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 13*n^3 + 39*n^2 - 60*n + 36 - (2*n^2 - 7*n + 6)*x^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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^4 - 5*n^3 + 11*n^2 - 12*n + (n^4 - 2*n^3 + n^2)*x - (3*n^2 - 4*n)*x^2 - (n^2 - n)*x^3)*v[i - 1] + ((4*n^3 - 12*n^2 + 16*n - 8)*x + (n^3 - 3*n^2 - 4*n + 6)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 13*n^3 + 39*n^2 - 60*n + 36 - (2*n^2 - 7*n + 6)*x^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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA …?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    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 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    ↑ 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