A143685 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(17\,n - 89\,n x + 36\,n x^{2} - 26\,n^{2} + 36\,n^{2} x + 9\,n^{3}\right)\,P_{n}(x) = \left(954\,x^{2} - 648\,x^{3} + 8\,n + 379\,n x - 1566\,n x^{2} + 648\,n x^{3} - 17\,n^{2} - 594\,n^{2} x + 648\,n^{2} x^{2} + 9\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(17\,x^{2} + 576\,x^{3} - 1296\,x^{4} + 8\,n x + 252\,n x^{2} - 1296\,n x^{3} - 9\,n^{2} x - 324\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	10	9			19
3	19	18			37
4	28	189	81		298
5	37	522	243		802
6	46	1017	2673	729	4465
7	55	1674	9558	2916	14203

All 15 rows

Technical details

Node id: n-f89a8fa507324261 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5425918f7a6aa7d9

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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:  1;
  1:  1;
  2: 10,    9;
  3: 19,   18;
  4: 28,  189,    81;
  5: 37,  522,   243;
  6: 46, 1017,  2673,    729;
  7: 55, 1674,  9558,   2916;
  8: 64, 2493, 23085,  33534,  6561;
  9: 73, 3474, 45441, 145800, 32805;

%F Formula

The row polynomials satisfy (9*n^3-26*n^2+17*n+(36*n^2-89*n)*x+36*n*x^2)*P(n,x) = (9*n^3-17*n^2+8*n+(162*n^3-594*n^2+379*n)*x+(648*n^2-1566*n+954)*x^2+(648*n-648)*x^3)*P(n-1,x) - ((9*n^2-8*n)*x+(324*n^2-252*n-17)*x^2+(1296*n-576)*x^3+1296*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 17*n^2 + 8*n + (162*n^3 - 594*n^2 + 379*n)*x + (648*n^2 - 1566*n + 954)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - ((9*n^2 - 8*n)*x + (324*n^2 - 252*n - 17)*x^2 + (1296*n - 576)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(9*n^3 - 26*n^2 + 17*n + (36*n^2 - 89*n)*x + 36*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^3 - 17*n^2 + 8*n + (162*n^3 - 594*n^2 + 379*n)*x + (648*n^2 - 1566*n + 954)*x^2 + (648*n - 648)*x^3)*v[i - 1] - ((9*n^2 - 8*n)*x + (324*n^2 - 252*n - 17)*x^2 + (1296*n - 576)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(9*n^3 - 26*n^2 + 17*n + (36*n^2 - 89*n)*x + 36*n*x^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-(1,9,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,9,1) array.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,9,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,9,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.