A122848 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 2\,x - n\right)\,P_{n-1}(x) + \left(1 + 2\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	8	14	4						26
3	22	66	48	8					144
4	66	272	348	144	16				846
5	206	1100	2020	1480	400	32			5238
6	688	4452	10680	11560	5520	1056	64		34020
7	2388	18284	54264	78120	56000	18816	2688	128	230688

All 15 rows

Technical details

Node id: n-98f59d4d16c87002 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-731283b4658cd846

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:     3,      2;
  2:     8,     14,       4;
  3:    22,     66,      48,       8;
  4:    66,    272,     348,     144,      16;
  5:   206,   1100,    2020,    1480,     400,      32;
  6:   688,   4452,   10680,   11560,    5520,    1056,     64;
  7:  2388,  18284,   54264,   78120,   56000,   18816,   2688,    128;
  8:  8732,  76736,  270928,  488768,  474320,  241024,  60032,   6656,   256;
  9: 33076, 329112, 1348848, 2927904, 3605616, 2513952, 948864, 182016, 16128, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(n-4-2*x)*P(n-1,x) + (1+2*x)*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[-(n - 4 - 2*x)*P[n - 1, x] + (1 + 2*x)*D[P[n - 1, x], x]];
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] = -(n - 4 - 2*x)*v[i - 1] + (1 + 2*x)*deriv(v[i - 1], 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_A013609 of Exponential Riordan array (1, x(1+x/2)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Exponential Riordan array (1, x(1+x/2)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).?
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
    
    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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
    
    ↑ 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 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…?
    
    ↑ 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 Exponential Riordan array (1, x(1+x/2)).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Exponential Riordan array (1, x(1+x/2)).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Parent-row provers of polytool polytool ? 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
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Exponential Riordan array (1, x(1+x/2)).?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Exponential Riordan array (1, x(1+x/2)).?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.