A105868 → D

nonnegative coefficients

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

Recurrence

\(\left(x + n - 2\,n x - n^{2} + n^{2} x\right)\,P_{n}(x) = \left(12 - 8\,x + 2\,x^{2} - 16\,n + 17\,n x - 6\,n x^{2} + 4\,n^{2} - 7\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(8\,x - 6\,x^{2} + x^{3} - 8\,n x + 10\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	2	2													4
3	0	12	3												15
4	0	12	36	4											52
5	0	0	90	80	5										175
6	0	0	60	360	150	6									576
7	0	0	0	560	1050	252	7								1869
8	0	0	0	280	2800	2520	392	8							6000
9	0	0	0	0	3150	10080	5292	576	9						19107
10	0	0	0	0	1260	18900	29400	10080	810	10					60460
11	0	0	0	0	0	16632	80850	73920	17820	1100	11				190333
12	0	0	0	0	0	5544	116424	277200	166320	29700	1452	12			596652
13	0	0	0	0	0	0	84084	576576	810810	343200	47190	1872	13		1863745
14	0	0	0	0	0	0	24024	672672	2270268	2102100	660660	72072	2366	14	5804176

Technical details

Node id: n-e6e255eef903e344 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-51897050898aff26

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: 1;
  2: 2,  2;
  3: 0, 12,  3;
  4: 0, 12, 36,   4;
  5: 0,  0, 90,  80,    5;
  6: 0,  0, 60, 360,  150,     6;
  7: 0,  0,  0, 560, 1050,   252,    7;
  8: 0,  0,  0, 280, 2800,  2520,  392,   8;
  9: 0,  0,  0,   0, 3150, 10080, 5292, 576, 9;

%F Formula

The row polynomials satisfy -(n^2-n-(n^2-2*n+1)*x)*P(n,x) = (4*n^2-16*n+12-(7*n^2-17*n+8)*x+(3*n^2-6*n+2)*x^2)*P(n-1,x) - ((8*n-8)*x-(10*n-6)*x^2+(2*n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 16*n + 12 - (7*n^2 - 17*n + 8)*x + (3*n^2 - 6*n + 2)*x^2)*P[n - 1, x] - ((8*n - 8)*x - (10*n - 6)*x^2 + (2*n - 1)*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (n^2 - 2*n + 1)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 16*n + 12 - (7*n^2 - 17*n + 8)*x + (3*n^2 - 6*n + 2)*x^2)*v[i - 1] - ((8*n - 8)*x - (10*n - 6)*x^2 + (2*n - 1)*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (n^2 - 2*n + 1)*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) = C(n,k)*C(k,n-k).?
  - 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.