A208509 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 - 2\,x + 14\,n - n x - 4\,n^{2} + n^{2} x\right)\,P_{n}(x) = \left(-16\,x + 2\,n + 21\,n x - 3\,n x^{2} - 4\,n^{2} - 3\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(8\,x - 8\,x^{2} - 8\,n x + 10\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

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

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	1			1
3	5			5
4	14	2		16
5	30	14		44
6	55	54	3	112
7	91	154	27	272

All 15 rows

Technical details

Node id: n-1e0f3b745edf2f86 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-579059b7d2bdb7eb

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 -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;
  4:  14,   2;
  5:  30,  14;
  6:  55,  54,   3;
  7:  91, 154,  27;
  8: 140, 364, 132,  4;
  9: 204, 756, 468, 44;

%F Formula

The row polynomials satisfy -(4*n^2-14*n+12-(n^2-n-2)*x)*P(n,x) = -(4*n^2-2*n+(3*n^2-21*n+16)*x-(n^2-3*n)*x^2)*P(n-1,x) - ((8*n-8)*x-(10*n-8)*x^2+2*n*x^3)*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[(-(4*n^2 - 2*n + (3*n^2 - 21*n + 16)*x - (n^2 - 3*n)*x^2)*P[n - 1, x] - ((8*n - 8)*x - (10*n - 8)*x^2 + 2*n*x^3)*D[P[n - 1, x], x])/(-4*n^2 + 14*n - 12 + (n^2 - 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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 2*n + (3*n^2 - 21*n + 16)*x - (n^2 - 3*n)*x^2)*v[i - 1] - ((8*n - 8)*x - (10*n - 8)*x^2 + 2*n*x^3)*deriv(v[i - 1], x))/(-4*n^2 + 14*n - 12 + (n^2 - 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 of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.?
  - 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.