A330381 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 9\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

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	6														9
2	9	33	27													69
3	27	141	225	108												501
4	81	546	1287	1242	405											3561
5	243	2004	6237	9072	6075	1458										25089
6	729	7107	27468	53487	54756	27459	5103									176109
7	2187	24603	113580	277371	383940	297675	117369	17496								1234221
8	6561	83652	449010	1319328	2307123	2437776	1500282	481140	59049							8643921
9	19683	280482	1716102	5895612	12474081	16749018	14141142	7138368	1909251	196830						60520569
10	59049	930021	6388353	25127118	62433585	101889171	109853739	76514382	32457267	7381125	649539					423683349
11	177147	3055785	23283747	103182498	294569379	566583903	745318665	665753418	391672017	142249041	27930177	2125764				2965901541
12	531441	9964518	83403009	411185754	1326432510	2940038046	4572312057	4999696326	3786523686	1916140050	604720809	103808142	6908733			20761665081
13	1594323	32285040	294451515	1598557788	5750940222	14439228642	25943140971	33628274280	31269607902	20440126278	9026643483	2505567168	379980315	22320522		145332718449
14	4782969	104029575	1026843642	6087075237	24164511219	67803877386	138292764561	207561703041	229340430783	184553752266	105609195549	41182957413	10155837510	1372712103	71744535	1017332217789

Technical details

Node id: n-53e8412f57bfc9da · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b57915851b3a9ea7

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,      6;
  2:     9,     33,      27;
  3:    27,    141,     225,     108;
  4:    81,    546,    1287,    1242,      405;
  5:   243,   2004,    6237,    9072,     6075,     1458;
  6:   729,   7107,   27468,   53487,    54756,    27459,     5103;
  7:  2187,  24603,  113580,  277371,   383940,   297675,   117369,   17496;
  8:  6561,  83652,  449010, 1319328,  2307123,  2437776,  1500282,  481140,   59049;
  9: 19683, 280482, 1716102, 5895612, 12474081, 16749018, 14141142, 7138368, 1909251, 196830;

%F Formula

The row polynomials satisfy P(n,x) = -(3+9*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 9*x + 9*x^2)*P[n - 2, x] + (4 + 6*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 9*x + 9*x^2)*v[i - 2] + (4 + 6*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