A245962 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-8\,n - 6\,n x + 3\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,n - 9\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	2			2
3	3			3
4	8	4		12
5	15	10		25
6	30	30	6	66
7	56	70	21	147

All 15 rows

Technical details

Node id: n-ef1acb736e5c4a3b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c12a5875baa63cc4

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:   2;
  3:   3;
  4:   8,   4;
  5:  15,  10;
  6:  30,  30,   6;
  7:  56,  70,  21;
  8: 104, 160,  72,  8;
  9: 189, 342, 198, 36;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (3*n^2-8*n+(2*n^2-6*n)*x)*P(n-1,x) - (5*n+9*n*x+4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + (2*n^2 - 6*n)*x)*P[n - 1, x] - (5*n + 9*n*x + 4*n*x^2)*D[P[n - 1, x], x])/(n^2 - 3*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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^2 - 8*n + (2*n^2 - 6*n)*x)*v[i - 1] - (5*n + 9*n*x + 4*n*x^2)*deriv(v[i - 1], x))/(n^2 - 3*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 induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.