A128908 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-4}(x) + \left(4 + 2\,x\right)\,P_{n-3}(x) + \left(-6 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)

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

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

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

\(P_{3}(x) = 3 + 8\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	2	2						4
3	3	8	3					14
4	4	20	18	4				46
5	5	40	63	32	5			145
6	6	70	168	144	50	6		444
7	7	112	378	480	275	72	7	1331

All 15 rows

Technical details

Node id: n-e9902435e1fd30c0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-832f8058e55c8435

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: 3,   8,    3;
  4: 4,  20,   18,    4;
  5: 5,  40,   63,   32,    5;
  6: 6,  70,  168,  144,   50,    6;
  7: 7, 112,  378,  480,  275,   72,   7;
  8: 8, 168,  756, 1320, 1100,  468,  98,   8;
  9: 9, 240, 1386, 3168, 3575, 2184, 735, 128, 9;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-4,x) + (4+2*x)*P(n-3,x) - (6+4*x+x^2)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 8*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 3 + 8*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-P[n - 4, x] + (4 + 2*x)*P[n - 3, x] - (6 + 4*x + x^2)*P[n - 2, x] + (4 + 2*x)*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] = 2 + 2*x; v[4] = 3 + 8*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -v[i - 4] + (4 + 2*x)*v[i - 3] - (6 + 4*x + x^2)*v[i - 2] + (4 + 2*x)*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 Riordan array (1, x/(1-x)^2).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.