A119808 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8\,x\right)\,P_{n-2}(x) + \left(3\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	9	20							29
3	27	84	32						143
4	81	324	256						661
5	243	1188	1440	256					3127
6	729	4212	6912	2816					14669
7	2187	14580	30240	19968	2048				69023
8	6561	49572	124416	115200	28672				324421
9	19683	166212	489888	587520	245760	16384			1525447
10	59049	551124	1866240	2757888	1658880	278528			7171709
11	177147	1810836	6928416	12192768	9676800	2801664	131072		33718703
12	531441	5904900	25194240	51508224	51093504	21676032	2621440		158529781
13	1594323	19131876	90069408	209952000	250822656	142442496	30277632	1048576	745338967
14	4782969	61647156	317447424	831409920	1164533760	836075520	264241152	24117248	3504255149

Technical details

Node id: n-3388161425ab794b · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-c5fb4726d5143de7

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,      4;
  2:     9,     20;
  3:    27,     84,     32;
  4:    81,    324,    256;
  5:   243,   1188,   1440,    256;
  6:   729,   4212,   6912,   2816;
  7:  2187,  14580,  30240,  19968,   2048;
  8:  6561,  49572, 124416, 115200,  28672;
  9: 19683, 166212, 489888, 587520, 245760, 16384;

%F Formula

The row polynomials satisfy P(n,x) = 8*x*P(n-2,x) + 3*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[8*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = 8*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