A122542 → D → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{3}\right)\,P_{n-6}(x) + \left(3\,x^{2} + 3\,x^{3}\right)\,P_{n-5}(x) + \left(3\,x + 3\,x^{2} + 3\,x^{3}\right)\,P_{n-4}(x) + \left(1 - 3\,x - 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 3\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)

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

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

\(P_{3}(x) = 2\)

\(P_{4}(x) = 8 + 6\,x\)

\(P_{5}(x) = 16 + 36\,x + 12\,x^{2}\)

\(P_{6}(x) = 24 + 108\,x + 96\,x^{2} + 20\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	P_n(1)
1														0
2														0
3	2													2
4	8	6												14
5	16	36	12											64
6	24	108	96	20										248
7	32	228	384	200	30									874
8	40	396	1056	1000	360	42								2894
9	48	612	2304	3400	2160	588	56							9168
10	56	876	4320	9000	8760	4116	896	72						28096
11	64	1188	7296	20040	27360	19404	7168	1296	90					83906
12	72	1548	11424	39400	70920	69972	38528	11664	1800	110				245438
13	80	1956	16896	70600	160080	207564	157696	70416	18000	2420	132			705840
14	88	2412	23904	117800	325080	530964	527744	322704	120600	26620	3168	156		2001240
15	96	2916	32640	185800	607680	1210188	1512448	1203984	612000	196020	38016	4056	182	5606026

Technical details

Node id: n-5a11f2c66162eaa4 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-037cdbea84276c0d

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
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:  0;
   3:  2;
   4:  8,   6;
   5: 16,  36,   12;
   6: 24, 108,   96,   20;
   7: 32, 228,  384,  200,   30;
   8: 40, 396, 1056, 1000,  360,   42;
   9: 48, 612, 2304, 3400, 2160,  588,  56;
  10: 56, 876, 4320, 9000, 8760, 4116, 896, 72;

%F Formula

The row polynomials satisfy P(n,x) = x^3*P(n-6,x) + (3*x^2+3*x^3)*P(n-5,x) + (3*x+3*x^2+3*x^3)*P(n-4,x) + (1-3*x-3*x^2+x^3)*P(n-3,x) - (3+3*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2, P(4,x) = 8 + 6*x, P(5,x) = 16 + 36*x + 12*x^2, P(6,x) = 24 + 108*x + 96*x^2 + 20*x^3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[4, x_] := 8 + 6*x;
P[5, x_] := 16 + 36*x + 12*x^2;
P[6, x_] := 24 + 108*x + 96*x^2 + 20*x^3;
P[n_, x_] := P[n, x] = Expand[x^3*P[n - 6, x] + (3*x^2 + 3*x^3)*P[n - 5, x] + (3*x + 3*x^2 + 3*x^3)*P[n - 4, x] + (1 - 3*x - 3*x^2 + x^3)*P[n - 3, x] - (3 + 3*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, 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] = 0; v[3] = 2; v[4] = 8 + 6*x; v[5] = 16 + 36*x + 12*x^2; v[6] = 24 + 108*x + 96*x^2 + 20*x^3; for(n = 7, N, my(i = n); v[i] = x^3*v[i - 6] + (3*x^2 + 3*x^3)*v[i - 5] + (3*x + 3*x^2 + 3*x^3)*v[i - 4] + (1 - 3*x - 3*x^2 + x^3)*v[i - 3] - (3 + 3*x + 3*x^2)*v[i - 2] + (3 + 3*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, 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 derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0,…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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.