A119900 → D

nonnegative coefficients

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

Recurrence

\(\left(2\,x + n - 6\,n x - n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(3 - 3\,x - 2\,x^{2} - 4\,n + 14\,n x - 14\,n x^{2} + n^{2} - 7\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} - 2\,n x + 10\,n x^{2} - 8\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	1	6													7
3	0	8	12												20
4	0	2	30	20											52
5	0	0	18	80	30										128
6	0	0	3	84	175	42									304
7	0	0	0	32	280	336	56								704
8	0	0	0	4	180	756	588	72							1600
9	0	0	0	0	50	720	1764	960	90						3584
10	0	0	0	0	5	330	2310	3696	1485	110					7936
11	0	0	0	0	0	72	1540	6336	7128	2200	132				17408
12	0	0	0	0	0	6	546	5720	15444	12870	3146	156			37888
13	0	0	0	0	0	0	98	2912	18018	34320	22022	4368	182		81920
14	0	0	0	0	0	0	7	840	12285	50050	70785	36036	5915	210	176128

Technical details

Node id: n-a6efc559e1790e92 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3e30a5fdddfb44b6

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: 2;
  2: 1, 6;
  3: 0, 8, 12;
  4: 0, 2, 30, 20;
  5: 0, 0, 18, 80,  30;
  6: 0, 0,  3, 84, 175,  42;
  7: 0, 0,  0, 32, 280, 336,   56;
  8: 0, 0,  0,  4, 180, 756,  588,  72;
  9: 0, 0,  0,  0,  50, 720, 1764, 960, 90;

%F Formula

The row polynomials satisfy -(n^2-n-(4*n^2-6*n+2)*x)*P(n,x) = (n^2-4*n+3-(7*n^2-14*n+3)*x+(12*n^2-14*n-2)*x^2)*P(n-1,x) - ((2*n-2)*x-(10*n-2)*x^2+8*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 4*n + 3 - (7*n^2 - 14*n + 3)*x + (12*n^2 - 14*n - 2)*x^2)*P[n - 1, x] - ((2*n - 2)*x - (10*n - 2)*x^2 + 8*n*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (4*n^2 - 6*n + 2)*x)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 4*n + 3 - (7*n^2 - 14*n + 3)*x + (12*n^2 - 14*n - 2)*x^2)*v[i - 1] - ((2*n - 2)*x - (10*n - 2)*x^2 + 8*n*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (4*n^2 - 6*n + 2)*x)); 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 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.