A081582 → γ → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-1 - 42\,x + n + 14\,n x\right)\,P_{n-1}(x) + \left(-x - 28\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 7\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	7			7
3	14			14
4	21	98		119
5	28	294		322
6	35	588	1029	1652
7	42	980	4116	5138

All 15 rows

Technical details

Node id: n-8d8a4edbb9b84147 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-983bc0c4ad0f2ee6

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:  7;
  3: 14;
  4: 21,   98;
  5: 28,  294;
  6: 35,  588,  1029;
  7: 42,  980,  4116;
  8: 49, 1470, 10290,  9604;
  9: 56, 2058, 20580, 48020;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-1+(14*n-42)*x)*P(n-1,x) - (x+28*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 7.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 7;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (14*n - 42)*x)*P[n - 1, x] - (x + 28*x^2)*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] = 7; for(n = 3, N, my(i = n + 1); v[i] = ((n - 1 + (14*n - 42)*x)*v[i - 1] - (x + 28*x^2)*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 gamma_polynomial of Pascal-(1,7,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,7,1) array.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,7,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,7,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.