A199881 → D+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 2 + x\)

\(P_{1}(x) = 1 + 3\,x + x^{2}\)

\(P_{2}(x) = 1 + 5\,x + 5\,x^{2} + x^{3}\)

\(P_{3}(x) = 4\,x + 11\,x^{2} + 7\,x^{3} + x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	1															3
1	1	3	1														5
2	1	5	5	1													12
3	0	4	11	7	1												23
4	0	2	13	20	9	1											45
5	0	0	9	31	32	11	1										84
6	0	0	3	29	62	47	13	1									155
7	0	0	0	16	74	110	65	15	1								281
8	0	0	0	4	56	161	179	86	17	1							504
9	0	0	0	0	25	155	312	273	110	19	1						895
10	0	0	0	0	5	97	366	554	396	137	21	1					1577
11	0	0	0	0	0	36	293	769	919	552	167	23	1				2760
12	0	0	0	0	0	6	155	750	1477	1444	745	200	25	1			4803
13	0	0	0	0	0	0	49	511	1701	2642	2171	979	236	27	1		8317
14	0	0	0	0	0	0	7	233	1415	3514	4461	3147	1258	275	29	1	14340

Technical details

Node id: n-70871ee425d37c88 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-03bc5cba23fc6c94

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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: 2, 1;
  1: 1, 3,  1;
  2: 1, 5,  5,  1;
  3: 0, 4, 11,  7,  1;
  4: 0, 2, 13, 20,  9,   1;
  5: 0, 0,  9, 31, 32,  11,   1;
  6: 0, 0,  3, 29, 62,  47,  13,   1;
  7: 0, 0,  0, 16, 74, 110,  65,  15,   1;
  8: 0, 0,  0,  4, 56, 161, 179,  86,  17,  1;
  9: 0, 0,  0,  0, 25, 155, 312, 273, 110, 19, 1;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - 2*x^2*P(n-3,x) + (2*x-x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 4, with P(0,x) = 2 + x, P(1,x) = 1 + 3*x + x^2, P(2,x) = 1 + 5*x + 5*x^2 + x^3, P(3,x) = 4*x + 11*x^2 + 7*x^3 + x^4.

%t Mathematica

P[0, x_] := 2 + x;
P[1, x_] := 1 + 3*x + x^2;
P[2, x_] := 1 + 5*x + 5*x^2 + x^3;
P[3, x_] := 4*x + 11*x^2 + 7*x^3 + x^4;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - 2*x^2*P[n - 3, x] + (2*x - x^2)*P[n - 2, x] + 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] = 2 + x; v[2] = 1 + 3*x + x^2; v[3] = 1 + 5*x + 5*x^2 + x^3; v[4] = 4*x + 11*x^2 + 7*x^3 + x^4; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - 2*x^2*v[i - 3] + (2*x - x^2)*v[i - 2] + 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 Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is …?
  - 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.