A059344 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 3\,n\right)\,P_{n}(x) = \left(-15\,n - 54\,n x + 6\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,n - 24\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	6			6
3	18			18
4	108	216		324
5	420	1080		1500
6	2250	9720	9720	21690
7	10206	52920	68040	131166

All 15 rows

Technical details

Node id: n-d2647fb576728480 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8f443bd722130c9e

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:      6;
  3:     18;
  4:    108,     216;
  5:    420,    1080;
  6:   2250,    9720,    9720;
  7:  10206,   52920,   68040;
  8:  55608,  378000,  816480,  544320;
  9: 281448, 2204496, 5715360, 4898880;

%F Formula

The row polynomials satisfy (3*n-6)*P(n,x) = (6*n^2-15*n+(18*n^2-54*n)*x)*P(n-1,x) - (4*n+24*n*x+36*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 15*n + (18*n^2 - 54*n)*x)*P[n - 1, x] - (4*n + 24*n*x + 36*n*x^2)*D[P[n - 1, x], x])/(3*n - 6)]];
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] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((6*n^2 - 15*n + (18*n^2 - 54*n)*x)*v[i - 1] - (4*n + 24*n*x + 36*n*x^2)*deriv(v[i - 1], x))/(3*n - 6)); 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 basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in t…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A059344 → basis A013610

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.