A129710 → D

nonnegative coefficients

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

Recurrence

\(\left(-10 - 4\,x + 9\,n - 2\,n^{2} + n^{2} x\right)\,P_{n}(x) = \left(-6 - 24\,x + 7\,n + 22\,n x - 6\,n x^{2} - 2\,n^{2} - 3\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 12\,x^{2} + 2\,n x + 7\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1								0
2	1							1
3	3							3
4	5	2						7
5	7	8						15
6	9	18	3					30
7	11	32	15					58
8	13	50	42	4				109
9	15	72	90	24				201
10	17	98	165	80	5			365
11	19	128	273	200	35			655
12	21	162	420	420	135	6		1164
13	23	200	612	784	385	48		2052
14	25	242	855	1344	910	210	7	3593

Technical details

Node id: n-fa81c8524277600a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-575fcd749d5c4035

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:  3;
  4:  5,  2;
  5:  7,  8;
  6:  9, 18,  3;
  7: 11, 32, 15;
  8: 13, 50, 42,  4;
  9: 15, 72, 90, 24;

%F Formula

The row polynomials satisfy -(2*n^2-9*n+10-(n^2-4)*x)*P(n,x) = -(2*n^2-7*n+6+(3*n^2-22*n+24)*x-(2*n^2-6*n)*x^2)*P(n-1,x) + ((2*n-3)*x+(7*n-12)*x^2-4*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[(-(2*n^2 - 7*n + 6 + (3*n^2 - 22*n + 24)*x - (2*n^2 - 6*n)*x^2)*P[n - 1, x] + ((2*n - 3)*x + (7*n - 12)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(-2*n^2 + 9*n - 10 + (n^2 - 4)*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] = (-(2*n^2 - 7*n + 6 + (3*n^2 - 22*n + 24)*x - (2*n^2 - 6*n)*x^2)*v[i - 1] + ((2*n - 3)*x + (7*n - 12)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(-2*n^2 + 9*n - 10 + (n^2 - 4)*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 Fibonacci binary words of length n and having k 01 subwords (0 <= k <= f…?
  - 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.