A104029 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

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	3														6
2	9	21	9													39
3	27	102	108	27												264
4	81	426	747	486	81											1821
5	243	1641	4059	4455	2025	243										12666
6	729	6015	19278	30159	23490	8019	729									88419
7	2187	21324	84069	171639	192213	113967	30618	2187								618204
8	6561	73812	345690	872856	1283283	1102248	520506	113724	6561							4325241
9	19683	250959	1361826	4102110	7492257	8492121	5848038	2270106	413343	19683						30270126
10	59049	841449	5192631	18185310	39764925	56210031	51313581	29222694	9546255	1476225	59049					211871199
11	177147	2790066	19297962	77075631	196519365	333604008	381110994	288900513	139217859	38972340	5196312	177147				1483039344
12	531441	9167358	70249869	315227646	918835002	1823350014	2508352677	2386887426	1536809274	637847298	155239821	18068994	531441			10381098261
13	1594323	29893557	251404821	1252390653	4109785452	9343243080	15059067453	17298529209	14020804512	7802262468	2828742345	605665593	62178597	1594323		72667156386
14	4782969	96855123	886941828	4857553557	17727683085	45455071158	84091005117	113402535057	111206804553	78107064750	38090915577	12203125389	2321334288	212044959	4782969	508668500379

Technical details

Node id: n-8ddb9ea84d7cbde5 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-35d2a8c113de1376

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,      3;
  2:     9,     21,       9;
  3:    27,    102,     108,      27;
  4:    81,    426,     747,     486,      81;
  5:   243,   1641,    4059,    4455,    2025,     243;
  6:   729,   6015,   19278,   30159,   23490,    8019,     729;
  7:  2187,  21324,   84069,  171639,  192213,  113967,   30618,    2187;
  8:  6561,  73812,  345690,  872856, 1283283, 1102248,  520506,  113724,   6561;
  9: 19683, 250959, 1361826, 4102110, 7492257, 8492121, 5848038, 2270106, 413343, 19683;

%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 + 3*x.

%t Mathematica

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