A104029 → D

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(8 + 6\,x - 6\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(1 - 3\,x - 3\,x^{2} - 4\,x^{3}\right)\,P_{n-3}'(x) + \left(-20 - 8\,x - 12\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 6\,x + 8\,x^{2}\right)\,P_{n-2}'(x) + \left(14 + 12\,x\right)\,P_{n-1}(x) + \left(1 - 4\,x\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	5	2													7
3	13	18	3												34
4	26	70	42	4											142
5	45	192	225	80	5										547
6	71	432	801	560	135	6									2005
7	105	854	2250	2492	1190	210	7								7108
8	148	1540	5400	8492	6440	2268	308	8							24604
9	201	2592	11574	24184	26055	14616	3990	432	9						83653
10	265	4134	22770	60460	86515	69030	30114	6600	585	10					280483
11	341	6314	41877	136840	248605	263538	163779	57552	10395	770	11				930022
12	430	9306	72930	286264	639270	860190	710710	356400	103554	15730	990	12			3055786
13	533	13312	121407	561704	1504750	2487504	2609243	1741168	723294	177320	23023	1248	13		9964519
14	651	18564	194571	1045044	3295110	6529698	8413419	7130760	3945942	1385670	291291	32760	1547	14	32285041

Technical details

Node id: n-f0a1fbef053ccf90 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-dfeb638fbefee7a9

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 -1)
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:   0;
  1:   1;
  2:   5,    2;
  3:  13,   18,     3;
  4:  26,   70,    42,     4;
  5:  45,  192,   225,    80,     5;
  6:  71,  432,   801,   560,   135,     6;
  7: 105,  854,  2250,  2492,  1190,   210,    7;
  8: 148, 1540,  5400,  8492,  6440,  2268,  308,   8;
  9: 201, 2592, 11574, 24184, 26055, 14616, 3990, 432, 9;

%F Formula

The row polynomials satisfy 4*P(n,x) = (8+6*x-6*x^2+4*x^3)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (20+8*x+12*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (14+12*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 6*x - 6*x^2 + 4*x^3)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (20 + 8*x + 12*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (14 + 12*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 6*x - 6*x^2 + 4*x^3)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (20 + 8*x + 12*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (14 + 12*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    ↑ 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.