A206306 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(4 - 2\,n\right)\,P_{n-2}(x) + \left(-8 - 6\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 4 + 15\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	4	15	9													28
3	14	66	81	27												188
4	48	276	504	351	81											1260
5	164	1116	2682	2862	1377	243										8444
6	560	4404	13068	18792	13932	5103	729									56588
7	1912	17064	60120	108648	109350	61722	18225	2187								379228
8	6528	65184	265536	577368	735480	564732	256608	63423	6561							2541420
9	22288	246192	1137456	2888784	4455324	4341924	2684178	1019142	216513	19683						17031484
10	76096	921264	4757328	13812768	25016688	29604204	23249268	12002256	3910356	728271	59049					114137548
11	259808	3420960	19518192	63745488	132594408	184783032	176441328	115718544	51215166	14604786	2421009	177147				764899868
12	887040	12620736	78820992	285910992	671580720	1077706944	1213615872	968193648	544195584	210608100	53380296	7971615	531441			5126023980
13	3028544	46302144	314109792	1252615968	3278867040	5956003872	7734701664	7282185120	4978932948	2445809580	840503466	191673054	26040609	1594323		34352368124
14	10340096	169052736	1237703616	5380971264	15530154624	31505202720	46379586528	50396458176	40673895984	24298820964	10592682012	3272259384	678118716	84499119	4782969	230214528908

Technical details

Node id: n-74ec83953eae1292 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-ff1dc72e140f0934

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:     1,      3;
  2:     4,     15,       9;
  3:    14,     66,      81,      27;
  4:    48,    276,     504,     351,      81;
  5:   164,   1116,    2682,    2862,    1377,     243;
  6:   560,   4404,   13068,   18792,   13932,    5103,     729;
  7:  1912,  17064,   60120,  108648,  109350,   61722,   18225,    2187;
  8:  6528,  65184,  265536,  577368,  735480,  564732,  256608,   63423,   6561;
  9: 22288, 246192, 1137456, 2888784, 4455324, 4341924, 2684178, 1019142, 216513, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4)*P(n-2,x) + (4*n-8+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 3*x, P(2,x) = 4 + 15*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[2, x_] := 4 + 15*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4)*P[n - 2, x] + (4*n - 8 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
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] = 1 + 3*x; v[3] = 4 + 15*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4)*v[i - 2] + (4*n - 8 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Riordan array (1, x/(1-3*x+2*x^2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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