A249253 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-4}(x) + \left(-2\,n x\right)\,P_{n-3}(x) + \left(2\,x - n - n^{2}\right)\,P_{n-2}(x) + \left(2 + 2\,n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 5\)

\(P_{3}(x) = 28 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5							5
3	28	2						30
4	180	22						202
5	1320	210	3					1533
6	10920	2040	57					13017
7	100800	21000	828	4				122632
8	1028160	231840	11340	116				1271456
9	11491200	2751840	156240	2380	5			14401665
10	139708800	35078400	2222640	43680	205			177053725
11	1836172800	479001600	33022080	776160	5640	6		2348978286
12	25945920000	6985440000	514926720	13829760	133560	330		33460250370
13	392302310400	108453945600	8442403200	251475840	2973600	11718	7	509453120365
14	6320426112000	1787154969600	145556611200	4710182400	64864800	347760	497	8257913088257

Technical details

Node id: n-01ad7c1c204c21a0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a2d8ef24ab47d92f

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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:        5;
  3:       28,       2;
  4:      180,      22;
  5:     1320,     210,      3;
  6:    10920,    2040,     57;
  7:   100800,   21000,    828,    4;
  8:  1028160,  231840,  11340,  116;
  9: 11491200, 2751840, 156240, 2380, 5;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*n*x*P(n-3,x) - (n^2+n-2*x)*P(n-2,x) + (2*n+2)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5, P(3,x) = 28 + 2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5;
P[3, x_] := 28 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*n*x*P[n - 3, x] - (n^2 + n - 2*x)*P[n - 2, x] + (2*n + 2)*P[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; v[3] = 5; v[4] = 28 + 2*x; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*n*x*v[i - 3] - (n^2 + n - 2*x)*v[i - 2] + (2*n + 2)*v[i - 1]); 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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.