A104029 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

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	28	16													53
3	27	136	192	64												419
4	81	568	1328	1152	256											3385
5	243	2188	7216	10560	6400	1024										27631
6	729	8020	34272	71488	74240	33792	4096									226637
7	2187	28432	149456	406848	607488	480256	172032	16384								1863083
8	6561	98416	614560	2068992	4055808	4644864	2924544	851968	65536							15331249
9	19683	334612	2421024	9723520	23679232	35785728	32858112	17006592	4128768	262144						126219415
10	59049	1121932	9231344	43105920	125676800	236868608	288313344	218923008	95354880	19660800	1048576					1039364261
11	177147	3720088	34307488	182697792	621098240	1405804544	2141331456	2164310016	1390608384	519045120	92274688	4194304				8559569267
12	531441	12223144	124888656	747206272	2903972352	7683581952	14093570048	17881464832	15350759424	8495038464	2756706304	427819008	16777216			70494539113
13	1594323	39858076	446941904	2968629696	12988951552	39372349440	84611715072	129592639488	140049907712	103912833024	50232033280	14340325376	1962934272	67108864		580587822079
14	4782969	129140164	1576785472	11514201024	56028232960	191547295744	472478404608	849559732224	1110813769728	1040252928000	676408066048	288933019648	73282879488	8925478912	268435456	4781723152445

Technical details

Node id: n-1ca301be4aef5c2e · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-a8c53caed96ace65

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,     28,      16;
  3:    27,    136,     192,      64;
  4:    81,    568,    1328,    1152,      256;
  5:   243,   2188,    7216,   10560,     6400,     1024;
  6:   729,   8020,   34272,   71488,    74240,    33792,     4096;
  7:  2187,  28432,  149456,  406848,   607488,   480256,   172032,    16384;
  8:  6561,  98416,  614560, 2068992,  4055808,  4644864,  2924544,   851968,   65536;
  9: 19683, 334612, 2421024, 9723520, 23679232, 35785728, 32858112, 17006592, 4128768, 262144;

%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 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*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 + 4*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, 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 [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 (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