A122542 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	9	12	3													24
4	27	63	39	7												136
5	81	270	288	120	17											776
6	243	1053	1620	1116	351	41										4424
7	729	3888	7857	7722	3933	996	99									25224
8	2187	13851	34749	44793	32211	13023	2763	239								143816
9	6561	48114	144342	231336	217242	123012	41256	7536	577							819976
10	19683	164025	572994	1100790	1283526	944946	441072	126432	20283	1393						4675144
11	59049	551124	2197935	4929498	6890508	6284952	3799386	1508220	377577	54012	3363					26655624
12	177147	1830519	8207811	21067371	34405884	37587240	28086426	14391594	4969539	1104471	142575	8119				151978696
13	531441	6022998	29996892	86762664	162338823	207174510	185179122	117030744	52015851	15893442	3176280	373608	19601			866515976
14	1594323	19663317	107705376	346735728	731951721	1070486199	1117053990	843296994	461370249	181012239	49598676	9005580	972951	47321		4940494664
15	4782969	63772920	381043197	1351631610	3179611503	5249141172	6275996721	5526710838	3606976089	1738855512	610516683	151636374	25227117	2520276	114243	28168537224

Technical details

Node id: n-c30f5d968923e55d · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-291062a4297424e9

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,      1;
   3:     9,     12,      3;
   4:    27,     63,     39,       7;
   5:    81,    270,    288,     120,      17;
   6:   243,   1053,   1620,    1116,     351,     41;
   7:   729,   3888,   7857,    7722,    3933,    996,     99;
   8:  2187,  13851,  34749,   44793,   32211,  13023,   2763,    239;
   9:  6561,  48114, 144342,  231336,  217242, 123012,  41256,   7536,   577;
  10: 19683, 164025, 572994, 1100790, 1283526, 944946, 441072, 126432, 20283, 1393;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(3*x + x^2)*P[n - 2, x] + (3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (3*x + x^2)*v[i - 2] + (3 + 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A122542 → basis A013610

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