A143683 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(15\,n - 79\,n x + 32\,n x^{2} - 23\,n^{2} + 32\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(752\,x^{2} - 512\,x^{3} + 7\,n + 289\,n x - 1232\,n x^{2} + 512\,n x^{3} - 15\,n^{2} - 464\,n^{2} x + 512\,n^{2} x^{2} + 8\,n^{3} + 128\,n^{3} x\right)\,P_{n-1}(x) + \left(15\,x^{2} + 448\,x^{3} - 1024\,x^{4} + 7\,n x + 192\,n x^{2} - 1024\,n x^{3} - 8\,n^{2} x - 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	9	8			17
3	17	16			33
4	25	152	64		241
5	33	416	192		641
6	41	808	1920	512	3281
7	49	1328	6784	2048	10209

All 15 rows

Technical details

Node id: n-c0041630a3f6a59b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-aa3b8de15dec955b

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:  9,    8;
  3: 17,   16;
  4: 25,  152,    64;
  5: 33,  416,   192;
  6: 41,  808,  1920,   512;
  7: 49, 1328,  6784,  2048;
  8: 57, 1976, 16320, 21504,  4096;
  9: 65, 2752, 32064, 92160, 20480;

%F Formula

The row polynomials satisfy (8*n^3-23*n^2+15*n+(32*n^2-79*n)*x+32*n*x^2)*P(n,x) = (8*n^3-15*n^2+7*n+(128*n^3-464*n^2+289*n)*x+(512*n^2-1232*n+752)*x^2+(512*n-512)*x^3)*P(n-1,x) - ((8*n^2-7*n)*x+(256*n^2-192*n-15)*x^2+(1024*n-448)*x^3+1024*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[((8*n^3 - 15*n^2 + 7*n + (128*n^3 - 464*n^2 + 289*n)*x + (512*n^2 - 1232*n + 752)*x^2 + (512*n - 512)*x^3)*P[n - 1, x] - ((8*n^2 - 7*n)*x + (256*n^2 - 192*n - 15)*x^2 + (1024*n - 448)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(8*n^3 - 23*n^2 + 15*n + (32*n^2 - 79*n)*x + 32*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] = ((8*n^3 - 15*n^2 + 7*n + (128*n^3 - 464*n^2 + 289*n)*x + (512*n^2 - 1232*n + 752)*x^2 + (512*n - 512)*x^3)*v[i - 1] - ((8*n^2 - 7*n)*x + (256*n^2 - 192*n - 15)*x^2 + (1024*n - 448)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(8*n^3 - 23*n^2 + 15*n + (32*n^2 - 79*n)*x + 32*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-(1,8,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,8,1) 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-(1,8,1) 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-(1,8,1) 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.