A078812 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 6\,n\right)\,P_{n}(x) = \left(9 + 9\,x + 9\,n + 9\,n x\right)\,P_{n-1}(x) + \left(5 + 18\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	3							3
3	18	18						36
4	75	162	81					318
5	270	918	972	324				2484
6	900	4212	6966	4860	1215			18153
7	2862	17118	38880	42120	21870	4374		127224
8	8817	64314	186705	277020	222345	91854	15309	866364

All 15 rows

Technical details

Node id: n-fdfe65b0aa25bc21 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-85a1027af8cb994c

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     0;
   2:     3;
   3:    18,     18;
   4:    75,    162,      81;
   5:   270,    918,     972,     324;
   6:   900,   4212,    6966,    4860,     1215;
   7:  2862,  17118,   38880,   42120,    21870,    4374;
   8:  8817,  64314,  186705,  277020,   222345,   91854,   15309;
   9: 26550, 228726,  810648, 1535760,  1683990, 1071630,  367416,   52488;
  10: 78585, 781164, 3274506, 7572852, 10588725, 9185400, 4837644, 1417176, 177147;

%F Formula

The row polynomials satisfy (6*n-12)*P(n,x) = (9*n+9+(9*n+9)*x)*P(n-1,x) + (5+18*x+9*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 9 + (9*n + 9)*x)*P[n - 1, x] + (5 + 18*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((9*n + 9 + (9*n + 9)*x)*v[i - 1] + (5 + 18*x + 9*x^2)*deriv(v[i - 1], x))/(6*n - 12)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023