A249248 → reverse → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	11	27	9													47
3	38	147	135	27												347
4	131	708	1161	594	81											2675
5	490	3375	7902	7263	2430	243										21703
6	1897	15927	48924	66771	39690	9477	729									183415
7	7714	76305	289845	528066	473445	198288	35721	2187								1611571
8	32501	370314	1678950	3847554	4691115	2973834	928746	131220	6561							14660795
9	142142	1832601	9656442	26707968	41501322	36207243	17084844	4142178	472392	19683						137766815
10	641795	9237615	55499499	180113706	341698095	384437421	251144145	91630926	17773749	1673055	59049					1333909055
11	2989294	47508939	320562783	1194853437	2683240020	3722503635	3164079429	1601454807	465227388	73909665	5845851	177147				13282352395
12	14321923	249104760	1865412855	7857353862	20417550120	33818639508	35753603067	23680971594	9539484048	2258427420	299437479	20194758	531441			135775032835
13	70493962	1331850783	10961394390	51502531293	152156366460	293701061106	373605389334	309934455705	163896168204	53709592590	10560205062	1186707753	69087330	1594323		1422686898295
14	355816769	7256241387	65109248154	337693132161	1118664502389	2469940569612	3686456785851	3704811791337	2466331685973	1062260883252	288369983979	47831638617	4615033644	234365481	4782969	15259936461575

Technical details

Node id: n-324bf3aa35ec200b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-dd3e37abd15c1cf3

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:      4,       3;
  2:     11,      27,       9;
  3:     38,     147,     135,       27;
  4:    131,     708,    1161,      594,       81;
  5:    490,    3375,    7902,     7263,     2430,      243;
  6:   1897,   15927,   48924,    66771,    39690,     9477,      729;
  7:   7714,   76305,  289845,   528066,   473445,   198288,    35721,    2187;
  8:  32501,  370314, 1678950,  3847554,  4691115,  2973834,   928746,  131220,   6561;
  9: 142142, 1832601, 9656442, 26707968, 41501322, 36207243, 17084844, 4142178, 472392, 19683;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (2 + 3*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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (2 + 3*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 reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    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 reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (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