A119808 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6\,x\right)\,P_{n-2}(x) + \left(3\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	3							6
2	9	15							24
3	27	63	18						108
4	81	243	144						468
5	243	891	810	108					2052
6	729	3159	3888	1188					8964
7	2187	10935	17010	8424	648				39204
8	6561	37179	69984	48600	9072				171396
9	19683	124659	275562	247860	77760	3888			749412
10	59049	413343	1049760	1163484	524880	66096			3276612
11	177147	1358127	3897234	5143824	3061800	664848	23328		14326308
12	531441	4428675	14171760	21730032	16166304	5143824	466560		62638596
13	1594323	14348907	50664042	88573500	79361856	33802272	5388768	139968	273873636
14	4782969	46235367	178564176	350751060	368465760	198404640	47029248	3219264	1197452484

Technical details

Node id: n-43f9a7f73cb3c4ce · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9a823f2771231d0b

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:     1;
  1:     3,      3;
  2:     9,     15;
  3:    27,     63,     18;
  4:    81,    243,    144;
  5:   243,    891,    810,    108;
  6:   729,   3159,   3888,   1188;
  7:  2187,  10935,  17010,   8424,   648;
  8:  6561,  37179,  69984,  48600,  9072;
  9: 19683, 124659, 275562, 247860, 77760, 3888;

%F Formula

The row polynomials satisfy P(n,x) = 6*x*P(n-2,x) + 3*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[6*x*P[n - 2, x] + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = 6*x*v[i - 2] + 3*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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceilin…?
  - 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