A107131 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	3							4
2	2	9	9						20
3	4	27	54	27					112
4	9	78	234	270	81				672
5	21	225	900	1620	1215	243			4224
6	51	648	3240	7965	9720	5103	729		27456
7	127	1869	11214	34965	59535	52731	20412	2187	183040

All 15 rows

Technical details

Node id: n-47a2e536b5967de9 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0a40c69c4c7d8789

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:   2,     9,      9;
  3:   4,    27,     54,     27;
  4:   9,    78,    234,    270,      81;
  5:  21,   225,    900,   1620,    1215,     243;
  6:  51,   648,   3240,   7965,    9720,    5103,     729;
  7: 127,  1869,  11214,  34965,   59535,   52731,   20412,    2187;
  8: 323,  5400,  37800, 142884,  312984,  394632,  265356,   78732,   6561;
  9: 835, 15633, 125064, 555660, 1490076, 2449440, 2388204, 1259712, 295245, 19683;

%F Formula

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

%t Mathematica

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

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from A Motzkin related triangle.?
  - 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