A330381 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

\(P_{2}(x) = 5 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	5	6													11
3	9	26	12												47
4	14	70	78	20											182
5	20	150	288	180	30										668
6	27	280	801	864	355	42									2369
7	35	476	1869	3000	2135	630	56								8201
8	44	756	3864	8492	9000	4620	1036	72							27884
9	54	1140	7308	20844	30230	23148	9072	1608	90						93494
10	65	1650	12906	46020	86515	90690	53130	16536	2385	110					310007
11	77	2310	21582	93588	219615	298326	239470	111672	28413	3410	132				1018595
12	90	3146	34518	178200	507650	860190	894978	572528	218790	46530	4730	156			3321506
13	104	4186	53196	321464	1088230	2236494	2902088	2407600	1263834	404690	73216	6396	182		10761680
14	119	5460	79443	554268	2192190	5348070	8413419	8706264	5931198	2612610	713427	111384	8463	210	34676525

Technical details

Node id: n-adce92670fb061df · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e96c84c692e18026

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:  2;
  2:  5,    6;
  3:  9,   26,   12;
  4: 14,   70,   78,    20;
  5: 20,  150,  288,   180,    30;
  6: 27,  280,  801,   864,   355,    42;
  7: 35,  476, 1869,  3000,  2135,   630,   56;
  8: 44,  756, 3864,  8492,  9000,  4620, 1036,   72;
  9: 54, 1140, 7308, 20844, 30230, 23148, 9072, 1608, 90;

%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) = 2, P(2,x) = 5 + 6*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 5 + 6*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] = 2; v[3] = 5 + 6*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: 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 [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 (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.