A104029 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-3 - 3\,x - x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	1							4
2	9	7	1						17
3	27	34	12	1					74
4	81	142	83	18	1				325
5	243	547	451	165	25	1			1432
6	729	2005	2142	1117	290	33	1		6317
7	2187	7108	9341	6357	2373	469	42	1	27878

All 15 rows

Technical details

Node id: n-fdc0c7cc2fd52cf9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-f2b7ac8f0752ccba

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:     3,     1;
  2:     9,     7,      1;
  3:    27,    34,     12,      1;
  4:    81,   142,     83,     18,     1;
  5:   243,   547,    451,    165,    25,     1;
  6:   729,  2005,   2142,   1117,   290,    33,    1;
  7:  2187,  7108,   9341,   6357,  2373,   469,   42,    1;
  8:  6561, 24604,  38410,  32328, 15843,  4536,  714,   52,  1;
  9: 19683, 83653, 151314, 151930, 92497, 34947, 8022, 1038, 63, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(3+3*x+x^2)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(3 + 3*x + x^2)*P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 3*x + x^2)*v[i - 2] + (4 + 2*x)*v[i - 1]); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (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 [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)
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)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A330381 → x ↦ x/(x+1)

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