A081579 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(7\,n - 39\,n x + 16\,n x^{2} - 11\,n^{2} + 16\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(184\,x^{2} - 128\,x^{3} + 3\,n + 49\,n x - 296\,n x^{2} + 128\,n x^{3} - 7\,n^{2} - 104\,n^{2} x + 128\,n^{2} x^{2} + 4\,n^{3} + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(7\,x^{2} + 96\,x^{3} - 256\,x^{4} + 3\,n x + 32\,n x^{2} - 256\,n x^{3} - 4\,n^{2} x - 64\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	4							9
3	9	8							17
4	13	44	16						73
5	17	112	48						177
6	21	212	288	64					585
7	25	344	928	256					1553
8	29	508	2160	1664	256				4617
9	33	704	4176	6400	1280				12593
10	37	932	7168	17600	8960	1024			35721
11	41	1192	11328	39424	39680	6144			97809
12	45	1484	16848	77056	125440	46080	4096		271049
13	49	1808	23920	136704	318976	229376	28672		739505
14	53	2164	32736	225600	698880	817152	229376	16384	2022345

Technical details

Node id: n-3e05b3ff0ad621f8 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-84c25a69d72b5faa

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:  5,   4;
  3:  9,   8;
  4: 13,  44,   16;
  5: 17, 112,   48;
  6: 21, 212,  288,   64;
  7: 25, 344,  928,  256;
  8: 29, 508, 2160, 1664,  256;
  9: 33, 704, 4176, 6400, 1280;

%F Formula

The row polynomials satisfy (4*n^3-11*n^2+7*n+(16*n^2-39*n)*x+16*n*x^2)*P(n,x) = (4*n^3-7*n^2+3*n+(32*n^3-104*n^2+49*n)*x+(128*n^2-296*n+184)*x^2+(128*n-128)*x^3)*P(n-1,x) - ((4*n^2-3*n)*x+(64*n^2-32*n-7)*x^2+(256*n-96)*x^3+256*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[((4*n^3 - 7*n^2 + 3*n + (32*n^3 - 104*n^2 + 49*n)*x + (128*n^2 - 296*n + 184)*x^2 + (128*n - 128)*x^3)*P[n - 1, x] - ((4*n^2 - 3*n)*x + (64*n^2 - 32*n - 7)*x^2 + (256*n - 96)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(4*n^3 - 11*n^2 + 7*n + (16*n^2 - 39*n)*x + 16*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] = ((4*n^3 - 7*n^2 + 3*n + (32*n^3 - 104*n^2 + 49*n)*x + (128*n^2 - 296*n + 184)*x^2 + (128*n - 128)*x^3)*v[i - 1] - ((4*n^2 - 3*n)*x + (64*n^2 - 32*n - 7)*x^2 + (256*n - 96)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(4*n^3 - 11*n^2 + 7*n + (16*n^2 - 39*n)*x + 16*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-(1,4,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,4,1) 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-(1,4,1) 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-(1,4,1) 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.