A085478 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-225 + 81\,x^{2} + 393\,n - 54\,n x^{2} - 243\,n^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(-39 + 99\,x + 189\,x^{2} + 81\,x^{3} + 126\,n - 18\,n x - 162\,n x^{2} - 81\,n x^{3} - 189\,n^{2} - 162\,n^{2} x + 81\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(-10 - 126\,x - 387\,x^{2} - 324\,x^{3} - 81\,x^{4} - 90\,n - 324\,n x - 162\,n x^{2} + 45\,n^{2} + 162\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	5	3													8
3	20	33	9												62
4	70	201	153	27											451
5	229	951	1269	621	81										3151
6	719	3924	7641	6534	2349	243									21410
7	2195	14868	38367	47898	29970	8505	729								142532
8	6565	53151	171423	284175	259200	127575	29889	2187							934165
9	19330	182145	706149	1466721	1776330	1272105	515403	102789	6561						6047533
10	56216	604473	2740077	6858378	10414008	9894717	5822523	2003292	347733	19683					38761100
11	161861	1956027	10156158	29792502	54568647	64996911	50693202	25290468	7558272	1161297	59049				246394394
12	462205	6201936	36304146	122241204	262700415	377536950	368957106	243714906	105461514	27851445	3838185	177147			1555447159
13	1310725	19337040	126014670	479263500	1183582125	1996670250	2354064930	1947239190	1114549875	425644875	100678545	12577437	531441		9761464603
14	3695045	59448459	426902994	1810557306	5056043085	9807328125	13574232234	13541799024	9699920181	4895358930	1672562925	358191234	40920957	1594323	60948554822

Technical details

Node id: n-c658d001ed5ce0ac · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-52d463599583deea

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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,      3;
  3:    20,     33,      9;
  4:    70,    201,    153,      27;
  5:   229,    951,   1269,     621,      81;
  6:   719,   3924,   7641,    6534,    2349,     243;
  7:  2195,  14868,  38367,   47898,   29970,    8505,    729;
  8:  6565,  53151, 171423,  284175,  259200,  127575,  29889,   2187;
  9: 19330, 182145, 706149, 1466721, 1776330, 1272105, 515403, 102789, 6561;

%F Formula

The row polynomials satisfy (54*n^3-243*n^2+393*n-225-(54*n-81)*x^2)*P(n,x) = (81*n^3-189*n^2+126*n-39+(81*n^3-162*n^2-18*n+99)*x-(162*n-189)*x^2-(81*n-81)*x^3)*P(n-1,x) + (45*n^2-90*n-10+(162*n^2-324*n-126)*x+(81*n^2-162*n-387)*x^2-324*x^3-81*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[((81*n^3 - 189*n^2 + 126*n - 39 + (81*n^3 - 162*n^2 - 18*n + 99)*x - (162*n - 189)*x^2 - (81*n - 81)*x^3)*P[n - 1, x] + (45*n^2 - 90*n - 10 + (162*n^2 - 324*n - 126)*x + (81*n^2 - 162*n - 387)*x^2 - 324*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 243*n^2 + 393*n - 225 - (54*n - 81)*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] = ((81*n^3 - 189*n^2 + 126*n - 39 + (81*n^3 - 162*n^2 - 18*n + 99)*x - (162*n - 189)*x^2 - (81*n - 81)*x^3)*v[i - 1] + (45*n^2 - 90*n - 10 + (162*n^2 - 324*n - 126)*x + (81*n^2 - 162*n - 387)*x^2 - 324*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 243*n^2 + 393*n - 225 - (54*n - 81)*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
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 T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    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 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    ↑ 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