A249247 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 16 + 12\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	5	3							8
3	16	12	1						29
4	66	56	7						129
5	336	306	48	1					691
6	2040	1944	351	13					4348
7	14400	14160	2808	140	1				31509
8	115920	116640	24720	1484	21				258785
9	1048320	1073520	239040	16300	330	1			2377511
10	10523520	10926720	2527560	189120	4805	31			24171756
11	116121600	121927680	29070720	2334360	69480	672	1		269524513
12	1397088000	1480550400	361791360	30710400	1027530	12900	43		3271180633
13	18202060800	19439481600	4848076800	430375680	15758400	237846	1232	1	42935992359
14	255307852800	274467916800	69634857600	6413904000	252292320	4370016	30191	57	606081223784

Technical details

Node id: n-e1e7ba0430b68730 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d813dac79b5b1484

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (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:       1;
  1:       2,       1;
  2:       5,       3;
  3:      16,      12,      1;
  4:      66,      56,      7;
  5:     336,     306,     48,     1;
  6:    2040,    1944,    351,    13;
  7:   14400,   14160,   2808,   140,   1;
  8:  115920,  116640,  24720,  1484,  21;
  9: 1048320, 1073520, 239040, 16300, 330, 1;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-4)*x*P(n-3,x) - (n^2-3*n+2-2*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 5 + 3*x, P(3,x) = 16 + 12*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 5 + 3*x;
P[3, x_] := 16 + 12*x + x^2;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 4)*x*P[n - 3, x] - (n^2 - 3*n + 2 - 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] = 1; v[2] = 2 + x; v[3] = 5 + 3*x; v[4] = 16 + 12*x + x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 4)*x*v[i - 3] - (n^2 - 3*n + 2 - 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.