A330381 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 12\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 8\,x\)

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	8														11
2	9	44	48													101
3	27	188	400	256												871
4	81	728	2288	2944	1280											7321
5	243	2672	11088	21504	19200	6144										60851
6	729	9476	48832	126784	173056	115712	28672									503261
7	2187	32804	201920	657472	1213440	1254400	659456	131072								4152751
8	6561	111536	798240	3127296	7291648	10272768	8429568	3604480	589824							34231921
9	19683	373976	3050848	13974784	39424256	70580224	79454208	53477376	19070976	2621440						282047771
10	59049	1240028	11357072	59560576	197320960	429360128	617230336	573210624	324206592	98304000	11534336					2323383701
11	177147	4074380	41393328	244580736	930984704	2387579904	4187688960	4987518976	3912302592	1894514688	495976448	50331648				19137123511
12	531441	13286024	148272016	974662528	4192181760	12389296128	25690247168	37455429632	37822529536	25519718400	10738466816	2457862144	218103808			157620587401
13	1594323	43046720	523469360	3789174016	18175811072	60846790656	145765576704	251927592960	312343396352	272227631104	160292667392	59324235776	11995709440	939524096		1298196219971
14	4782969	138706100	1825499808	14428622784	76371788544	285724981248	777019428864	1554956992512	2290817630208	2457941311488	1875379224576	975087599616	320612597760	57780731904	4026531840	10692116430221

Technical details

Node id: n-80d93a39aa9a55a2 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-eb297539291db79c

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,      8;
  2:     9,     44,      48;
  3:    27,    188,     400,      256;
  4:    81,    728,    2288,     2944,     1280;
  5:   243,   2672,   11088,    21504,    19200,     6144;
  6:   729,   9476,   48832,   126784,   173056,   115712,    28672;
  7:  2187,  32804,  201920,   657472,  1213440,  1254400,   659456,   131072;
  8:  6561, 111536,  798240,  3127296,  7291648, 10272768,  8429568,  3604480,   589824;
  9: 19683, 373976, 3050848, 13974784, 39424256, 70580224, 79454208, 53477376, 19070976, 2621440;

%F Formula

The row polynomials satisfy P(n,x) = -(3+12*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 12*x + 16*x^2)*P[n - 2, x] + (4 + 8*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 12*x + 16*x^2)*v[i - 2] + (4 + 8*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