A249252 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(20 + 28\,x + 8\,x^{2} - 7\,n - 6\,n x\right)\,P_{n-1}(x) + \left(17 + 37\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	4							4
2	24	16						40
3	124	160	48					332
4	576	1104	672	128				2480
5	2532	6432	5904	2304	320			17492
6	10728	33936	41664	24704	7040	768		118840
7	44300	167712	257904	205568	89280	19968	1792	786524

All 15 rows

Technical details

Node id: n-d24be80d81a1d6e9 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-01f0f465f1c5e3f5

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:      4;
  2:     24,      16;
  3:    124,     160,      48;
  4:    576,    1104,     672,     128;
  5:   2532,    6432,    5904,    2304,     320;
  6:  10728,   33936,   41664,   24704,    7040,     768;
  7:  44300,  167712,  257904,  205568,   89280,   19968,   1792;
  8: 179472,  791120, 1461984, 1464960,  857600,  292608,  53760,   4096;
  9: 716468, 3603872, 7778256, 9392384, 6927680, 3190272, 894208, 139264, 9216;

%F Formula

The row polynomials satisfy (n-1+(2*n-2)*x)*P(n,x) = -(7*n-20+(6*n-28)*x-8*x^2)*P(n-1,x) + (17+37*x+24*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(7*n - 20 + (6*n - 28)*x - 8*x^2)*P[n - 1, x] + (17 + 37*x + 24*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(7*n - 20 + (6*n - 28)*x - 8*x^2)*v[i - 1] + (17 + 37*x + 24*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x)); 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 basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    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 reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ 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.