A103327 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 7\,x - 2\,x^{2} + 7\,n - 18\,n x + 4\,n x^{2} - 8\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-1 + 14\,x + 13\,x^{2} - 6\,x^{3} + n - 27\,n x - 38\,n x^{2} + 12\,n x^{3} - 8\,n^{2} x + 24\,n^{2} x^{2} + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2} + 20\,x^{3} - 8\,x^{4} + 16\,n x^{2} - 16\,n x^{3} + 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	11	20	5													36
3	22	91	56	7												176
4	37	288	378	120	9											832
5	56	715	1716	1122	220	11										3840
6	79	1508	5863	6864	2717	364	13									17408
7	106	2835	16380	30745	21450	5733	560	15								77824
8	137	4896	39508	109616	121550	56576	10948	816	17							344064
9	172	7923	85272	329460	537472	394706	131784	19380	1140	19						1507328
10	211	12180	168777	868224	1967070	2116296	1107890	279072	32319	1540	21					6553600
11	254	17963	311696	2062203	6210644	9256534	7072408	2778446	547998	51359	2024	23				28311552
12	301	25600	543950	4503400	17425375	34470560	36402100	20801200	6369275	1012000	78430	2600	25			121634816
13	352	35451	905580	9176310	44401500	112739445	157791960	124361460	55219320	13567125	1776060	115830	3276	27		520093696
14	407	47908	1448811	17643600	104442195	331405620	594807255	620470080	379239525	134747340	27183585	2985840	166257	4060	29	2214592512

Technical details

Node id: n-9210bfff0db445ed · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5e0ca6f4c7830ccb

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:   4,    3;
  2:  11,   20,     5;
  3:  22,   91,    56,      7;
  4:  37,  288,   378,    120,      9;
  5:  56,  715,  1716,   1122,    220,     11;
  6:  79, 1508,  5863,   6864,   2717,    364,     13;
  7: 106, 2835, 16380,  30745,  21450,   5733,    560,    15;
  8: 137, 4896, 39508, 109616, 121550,  56576,  10948,   816,   17;
  9: 172, 7923, 85272, 329460, 537472, 394706, 131784, 19380, 1140, 19;

%F Formula

The row polynomials satisfy (4*n^3-8*n^2+7*n-2+(8*n^2-18*n+7)*x+(4*n-2)*x^2)*P(n,x) = (4*n^3+n-1+(12*n^3-8*n^2-27*n+14)*x+(24*n^2-38*n+13)*x^2+(12*n-6)*x^3)*P(n-1,x) + ((8*n^2-6)*x-(8*n^2-16*n+6)*x^2-(16*n-20)*x^3-8*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[((4*n^3 + n - 1 + (12*n^3 - 8*n^2 - 27*n + 14)*x + (24*n^2 - 38*n + 13)*x^2 + (12*n - 6)*x^3)*P[n - 1, x] + ((8*n^2 - 6)*x - (8*n^2 - 16*n + 6)*x^2 - (16*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 8*n^2 + 7*n - 2 + (8*n^2 - 18*n + 7)*x + (4*n - 2)*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] = ((4*n^3 + n - 1 + (12*n^3 - 8*n^2 - 27*n + 14)*x + (24*n^2 - 38*n + 13)*x^2 + (12*n - 6)*x^3)*v[i - 1] + ((8*n^2 - 6)*x - (8*n^2 - 16*n + 6)*x^2 - (16*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 8*n^2 + 7*n - 2 + (8*n^2 - 18*n + 7)*x + (4*n - 2)*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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

Related ?

Same fingerprint ?

A091042 → D+1

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