A128908 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 3 + 12\,x + 9\,x^{2}\)

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	3	12	9													24
3	8	42	63	27												140
4	21	138	306	270	81											816
5	55	435	1269	1701	1053	243										4756
6	144	1332	4806	8640	8181	3888	729									27720
7	377	3993	17145	38637	49410	35964	13851	2187								161564
8	987	11778	58608	158706	255960	252234	148716	48114	6561							941664
9	2584	34302	194013	613305	1194588	1488618	1188999	588303	164025	19683						5488420
10	6765	98880	626337	2263248	5167719	7797384	7884135	5283792	2250423	551124	59049					31988856
11	17711	282633	1981638	8055450	21098313	37406691	45855558	38915478	22438620	8384958	1830519	177147				186444716
12	46368	802152	6166476	27848016	82293570	167717628	241902612	249029316	181811871	91919610	30587382	6022998	531441			1086679440
13	121393	2262927	18924246	93988026	309326445	712626903	1183005162	1433880306	1270084941	812809485	365690457	109653993	19663317	1594323		6333631924
14	317811	6350808	57395043	310888800	1127649843	2898142416	5444993583	7601627088	7930083870	6156763668	3504912444	1420010352	387420489	63772920	4782969	36915112104

Technical details

Node id: n-a613aa7e2e807e74 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-d58b2b7656047d39

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:    3,    12,      9;
  3:    8,    42,     63,     27;
  4:   21,   138,    306,    270,      81;
  5:   55,   435,   1269,   1701,    1053,     243;
  6:  144,  1332,   4806,   8640,    8181,    3888,     729;
  7:  377,  3993,  17145,  38637,   49410,   35964,   13851,   2187;
  8:  987, 11778,  58608, 158706,  255960,  252234,  148716,  48114,   6561;
  9: 2584, 34302, 194013, 613305, 1194588, 1488618, 1188999, 588303, 164025, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2)*P(n-2,x) + (3*n-6+(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) = 3 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[2, x_] := 3 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*P[n - 2, x] + (3*n - 6 + (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] = 3 + 12*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2)*v[i - 2] + (3*n - 6 + (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-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