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	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

All 15 rows

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.