A098158 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 20\,x + 3\,n - 18\,n x - n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(4 - 20\,x + 34\,x^{2} - 5\,n + 27\,n x - 46\,n x^{2} + n^{2} - 7\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 10\,x^{2} + 8\,x^{3} - 2\,n x + 10\,n x^{2} - 8\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	1	2													3
4	0	6	3												9
5	0	2	18	4											24
6	0	0	15	40	5										60
7	0	0	3	60	75	6									144
8	0	0	0	28	175	126	7								336
9	0	0	0	4	140	420	196	8							768
10	0	0	0	0	45	504	882	288	9						1728
11	0	0	0	0	5	270	1470	1680	405	10					3840
12	0	0	0	0	0	66	1155	3696	2970	550	11				8448
13	0	0	0	0	0	6	462	3960	8316	4950	726	12			18432
14	0	0	0	0	0	0	91	2288	11583	17160	7865	936	13		39936
15	0	0	0	0	0	0	7	728	9009	30030	33033	12012	1183	14	86016

Technical details

Node id: n-bdbabe9e8ea4ed15 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-40fd595992edbfb8

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0;
   2: 1;
   3: 1, 2;
   4: 0, 6,  3;
   5: 0, 2, 18,  4;
   6: 0, 0, 15, 40,   5;
   7: 0, 0,  3, 60,  75,   6;
   8: 0, 0,  0, 28, 175, 126,   7;
   9: 0, 0,  0,  4, 140, 420, 196,   8;
  10: 0, 0,  0,  0,  45, 504, 882, 288, 9;

%F Formula

The row polynomials satisfy -(n^2-3*n+2-(4*n^2-18*n+20)*x)*P(n,x) = (n^2-5*n+4-(7*n^2-27*n+20)*x+(12*n^2-46*n+34)*x^2)*P(n-1,x) - ((2*n-2)*x-(10*n-10)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 5*n + 4 - (7*n^2 - 27*n + 20)*x + (12*n^2 - 46*n + 34)*x^2)*P[n - 1, x] - ((2*n - 2)*x - (10*n - 10)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(-n^2 + 3*n - 2 + (4*n^2 - 18*n + 20)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - 5*n + 4 - (7*n^2 - 27*n + 20)*x + (12*n^2 - 46*n + 34)*x^2)*v[i - 1] - ((2*n - 2)*x - (10*n - 10)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(-n^2 + 3*n - 2 + (4*n^2 - 18*n + 20)*x)); 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) with diagonals T(n,n-k) = binomial(n, 2*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.