A081580 → γ → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-1 - 30\,x + n + 10\,n x\right)\,P_{n-1}(x) + \left(-x - 20\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 5\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1								0
2	5							5
3	10							10
4	15	50						65
5	20	150						170
6	25	300	375					700
7	30	500	1500					2030
8	35	750	3750	2500				7035
9	40	1050	7500	12500				21090
10	45	1400	13125	37500	15625			67695
11	50	1800	21000	87500	93750			204100
12	55	2250	31500	175000	328125	93750		630680
13	60	2750	45000	315000	875000	656250		1894060
14	65	3300	61875	525000	1968750	2625000	546875	5730865

Technical details

Node id: n-92233194929d4d10 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-4fabd50f522b7949

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:  5;
  3: 10;
  4: 15,   50;
  5: 20,  150;
  6: 25,  300,  375;
  7: 30,  500, 1500;
  8: 35,  750, 3750,  2500;
  9: 40, 1050, 7500, 12500;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-1+(10*n-30)*x)*P(n-1,x) - (x+20*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 5.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (10*n - 30)*x)*P[n - 1, x] - (x + 20*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] = 5; for(n = 3, N, my(i = n + 1); v[i] = ((n - 1 + (10*n - 30)*x)*v[i - 1] - (x + 20*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-(1,5,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,5,1) 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-(1,5,1) 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-(1,5,1) 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.