A008306 → D

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 20\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	3			3
3	20			20
4	130	30		160
5	924	420		1344
6	7308	4760	315	12383
7	64224	52864	7560	124648

All 15 rows

Technical details

Node id: n-02f5dc55726c7d59 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-626d2a356d5551c8

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
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:       0;
  2:       3;
  3:      20;
  4:     130,      30;
  5:     924,     420;
  6:    7308,    4760,     315;
  7:   64224,   52864,    7560;
  8:  623376,  607320,  132300,   3780;
  9: 6636960, 7357680, 2115960, 138600;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n^3-n)*x^2*P(n-4,x) - (2*n^3-2*n)*x*P(n-3,x) - (n^3-n-(2*n^2+2*n)*x)*P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3, P(3,x) = 20.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 3;
P[3, x_] := 20;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n)*x^2*P[n - 4, x] - (2*n^3 - 2*n)*x*P[n - 3, x] - (n^3 - n - (2*n^2 + 2*n)*x)*P[n - 2, x] + (2*n^2 + n - 1)*P[n - 1, x])/(n)]];
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] = 0; v[3] = 3; v[4] = 20; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - n)*x^2*v[i - 4] - (2*n^3 - 2*n)*x*v[i - 3] - (n^3 - n - (2*n^2 + 2*n)*x)*v[i - 2] + (2*n^2 + n - 1)*v[i - 1])/(n)); 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: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows; related to number of preorders.?
    
    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.