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	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

All 15 rows

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.