A136394 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 6\,x - 3\,x^{2} + 4\,n - 8\,n x + 4\,n x^{2} - n^{2} + 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(4 - 4\,x - 6\,n + 6\,n x + 2\,n^{2} - 2\,n^{2} x\right)\,P_{n-3}(x) + \left(1 - 2\,x + 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 2\,n\right)\,P_{n-1}(x)\)

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

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

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

\(P_{3}(x) = 5\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	1			1
3	5			5
4	20	6		26
5	84	70		154
6	409	590	45	1044
7	2365	4718	945	8028

All 15 rows

Technical details

Node id: n-d68e6fcf9b7ad391 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7a2ba97e1e8344f1

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:      5;
  4:     20,      6;
  5:     84,     70;
  6:    409,    590,     45;
  7:   2365,   4718,    945;
  8:  16064,  39340,  13440,   420;
  9: 125664, 354156, 170016, 13860;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-4,x) + (2*n^2-6*n+4-(2*n^2-6*n+4)*x)*P(n-3,x) - (n^2-1-(2*n-2)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 5.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 5;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*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] = 0; v[2] = 0; v[3] = 1; v[4] = 5; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 4] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*v[i - 3] - (n^2 - 1 - (2*n - 2)*x)*v[i - 2] + (2*n - 1)*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 Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles 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.