A124324 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(6\,x^{2} - 11\,n x^{2} + 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-4\,x + 6\,n x - 2\,n^{2} x\right)\,P_{n-3}(x) + \left(-2\,x^{2} + 3\,n x^{2} - n^{2} x^{2}\right)\,P_{n-3}'(x) + \left(2 + 4\,x - 2\,n - 6\,n x + 2\,n^{2} x\right)\,P_{n-2}(x) + \left(x - n x\right)\,P_{n-2}'(x) + \left(-5 + 3\,n\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-1}'(x)\)

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

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

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

\(P_{3}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	1			1
3	4			4
4	11	6		17
5	26	50		76
6	57	260	45	362
7	120	1092	630	1842

All 15 rows

Technical details

Node id: n-7368f216018f4712 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a87fa6ef6c203402

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:   1;
  3:   4;
  4:  11,     6;
  5:  26,    50;
  6:  57,   260,    45;
  7: 120,  1092,   630;
  8: 247,  4074,  5250,  420;
  9: 502, 14142, 34104, 8820;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n^3-6*n^2+11*n-6)*x^2*P(n-4,x) - (2*n^2-6*n+4)*x*P(n-3,x) - (n^2-3*n+2)*x^2*d/dx P(n-3,x) - (2*n-2-(2*n^2-6*n+4)*x)*P(n-2,x) - (n-1)*x*d/dx P(n-2,x) + (3*n-5)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 6*n^2 + 11*n - 6)*x^2*P[n - 4, x] - (2*n^2 - 6*n + 4)*x*P[n - 3, x] - (n^2 - 3*n + 2)*x^2*D[P[n - 3, x], x] - (2*n - 2 - (2*n^2 - 6*n + 4)*x)*P[n - 2, x] - (n - 1)*x*D[P[n - 2, x], x] + (3*n - 5)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];
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] = 1; v[4] = 4; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - 6*n^2 + 11*n - 6)*x^2*v[i - 4] - (2*n^2 - 6*n + 4)*x*v[i - 3] - (n^2 - 3*n + 2)*x^2*deriv(v[i - 3], x) - (2*n - 2 - (2*n^2 - 6*n + 4)*x)*v[i - 2] - (n - 1)*x*deriv(v[i - 2], x) + (3*n - 5)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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 read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=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

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.