A201730 → D

nonnegative coefficients

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

Recurrence

\(\left(12 - 10\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(12 + 4\,x - 17\,n - 5\,n x + 5\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-6 - 4\,x + 2\,x^{2} + 6\,n + 4\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 0\)

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

Rows ?

n	x⁰	x¹	x²	P_n(1)
1				0
2				0
3	1			1
4	6			6
5	26	2		28
6	100	20		120
7	363	126	3	492
8	1274	644	42	1960

All 15 rows

Technical details

Node id: n-ea3e2b3e5b9696f4 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-82831e338d4d8cff

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -3)
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:     0;
   3:     1;
   4:     6;
   5:    26,     2;
   6:   100,    20;
   7:   363,   126,    3;
   8:  1274,   644,   42;
   9:  4372,  2924,  348,  4;
  10: 14760, 12312, 2232, 72;

%F Formula

The row polynomials satisfy (2*n^2-10*n+12)*P(n,x) = (5*n^2-17*n+12+(n^2-5*n+4)*x)*P(n-1,x) + (6*n-6+(4*n-4)*x-(2*n-2)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 17*n + 12 + (n^2 - 5*n + 4)*x)*P[n - 1, x] + (6*n - 6 + (4*n - 4)*x - (2*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 10*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] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = ((5*n^2 - 17*n + 12 + (n^2 - 5*n + 4)*x)*v[i - 1] + (6*n - 6 + (4*n - 4)*x - (2*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 10*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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.