A119808: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).

A119808 ↗ nonnegative coefficients

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

Recurrence from A119808

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	4	5							9
3	8	17	2						27
4	16	49	16						81
5	32	129	78	4					243
6	64	321	300	44					729
7	128	769	1002	280	8				2187
8	256	1793	3048	1352	112				6561
9	512	4097	8678	5500	880	16			19683
10	1024	9217	23524	19892	5120	272			59049
11	2048	20481	61410	66032	24600	2544	32		177147
12	4096	45057	155616	205360	103344	17328	640		531441
13	8192	98305	384990	606836	392896	96096	6944	64	1594323
14	16384	212993	933852	1721020	1382720	460320	54208	1472	4782969

b-files: A000244 (row sums), A119808 ?

Technical details

Node id: n-6120ac587df8130e · revision 0 · JSON
Definition source: oeis_catalog:A119808
Fingerprint: f6-d6c40c3a1672151f

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 ?

A119808 triangle 52 terms, 12 rows · tabf · nonn · catalog b-file?

Row sums ? A354269, A155734, A000244, A140429, A027313, A376478, A133494

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   2,    1;
  2:   4,    5;
  3:   8,   17,    2;
  4:  16,   49,   16;
  5:  32,  129,   78,    4;
  6:  64,  321,  300,   44;
  7: 128,  769, 1002,  280,   8;
  8: 256, 1793, 3048, 1352, 112;
  9: 512, 4097, 8678, 5500, 880, 16;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (2 - 2*x)*P[n - 2, 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (2 - 2*x)*v[i - 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

A119808 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,    1;
  2:   4,    5;
  3:   8,   17,    2;
  4:  16,   49,   16;
  5:  32,  129,   78,    4;
  6:  64,  321,  300,   44;
  7: 128,  769, 1002,  280,   8;
  8: 256, 1793, 3048, 1352, 112;
  9: 512, 4097, 8678, 5500, 880, 16;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (2 - 2*x)*P[n - 2, 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (2 - 2*x)*v[i - 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 Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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