A330381 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 6\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	9	22	12													43
3	27	94	100	32												253
4	81	364	572	368	80											1465
5	243	1336	2772	2688	1200	192										8431
6	729	4738	12208	15848	10816	3616	448									48403
7	2187	16402	50480	82184	75840	39200	10304	1024								277621
8	6561	55768	199560	390912	455728	321024	131712	28160	2304							1591729
9	19683	186988	762712	1746848	2464016	2205632	1241472	417792	74496	5120						9124759
10	59049	620014	2839268	7445072	12332560	13417504	9644224	4478208	1266432	192000	11264					52305595
11	177147	2037190	10348332	30572592	58186544	74611872	65432640	38964992	15282432	3700224	484352	24576				299822893
12	531441	6643012	37068004	121832816	262011360	387165504	401410112	292620544	147744256	49843200	10486784	1200128	53248			1718610409
13	1594323	21523360	130867340	473646752	1135988192	1901462208	2277587136	1968184320	1220091392	531694592	156535808	28966912	2928640	114688		9851185663
14	4782969	69353050	456374952	1803577848	4773236784	8928905664	12140928576	12148101504	8948506368	4800666624	1831425024	476116992	78274560	7053312	245760	56467549987

Technical details

Node id: n-703053a921044202 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-0aeeb4be61492791

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,      4;
  2:     9,     22,     12;
  3:    27,     94,    100,      32;
  4:    81,    364,    572,     368,      80;
  5:   243,   1336,   2772,    2688,    1200,     192;
  6:   729,   4738,  12208,   15848,   10816,    3616,     448;
  7:  2187,  16402,  50480,   82184,   75840,   39200,   10304,   1024;
  8:  6561,  55768, 199560,  390912,  455728,  321024,  131712,  28160,  2304;
  9: 19683, 186988, 762712, 1746848, 2464016, 2205632, 1241472, 417792, 74496, 5120;

%F Formula

The row polynomials satisfy P(n,x) = -(3+6*x+4*x^2)*P(n-2,x) + (4+4*x)*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[-(3 + 6*x + 4*x^2)*P[n - 2, x] + (4 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 6*x + 4*x^2)*v[i - 2] + (4 + 4*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 (−∞, 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 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 [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 (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