A078812 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-16 + 8\,n\right)\,P_{n}(x) = \left(12 + 16\,x + 12\,n + 16\,n x\right)\,P_{n-1}(x) + \left(5 + 24\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	4							4
3	24	32						56
4	100	288	192					580
5	360	1632	2304	1024				5320
6	1200	7488	16512	15360	5120			45680
7	3816	30432	92160	133120	92160	24576		376264
8	11756	114336	442560	875520	936960	516096	114688	3011916

All 15 rows

Technical details

Node id: n-3e9edb6397a669eb · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6d3434c9512a4a8b

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:      4;
   3:     24,      32;
   4:    100,     288,     192;
   5:    360,    1632,    2304,     1024;
   6:   1200,    7488,   16512,    15360,     5120;
   7:   3816,   30432,   92160,   133120,    92160,    24576;
   8:  11756,  114336,  442560,   875520,   936960,   516096,   114688;
   9:  35400,  406624, 1921536,  4853760,  7096320,  6021120,  2752512,   524288;
  10: 104780, 1388736, 7761792, 23933952, 44620800, 51609600, 36241408, 14155776, 2359296;

%F Formula

The row polynomials satisfy (8*n-16)*P(n,x) = (12*n+12+(16*n+16)*x)*P(n-1,x) + (5+24*x+16*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 12 + (16*n + 16)*x)*P[n - 1, x] + (5 + 24*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((12*n + 12 + (16*n + 16)*x)*v[i - 1] + (5 + 24*x + 16*x^2)*deriv(v[i - 1], x))/(8*n - 16)); 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_A013611 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