A201730 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	2				2
3	6	1			7
4	20	6			26
5	68	29	1		98
6	232	130	10		372
7	792	559	69	1	1421
8	2704	2338	406	14	5462

All 15 rows

Technical details

Node id: n-34bf3ddff9651c72 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-4b0c268675a7a0d3

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     2;
   3:     6,     1;
   4:    20,     6;
   5:    68,    29,     1;
   6:   232,   130,    10;
   7:   792,   559,    69,   1;
   8:  2704,  2338,   406,  14;
   9:  9232,  9585,  2183, 126,  1;
  10: 31520, 38706, 11070, 924, 18;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 2, x] + x*D[P[n - 2, x], x] + 4*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] = 2; for(n = 3, N, my(i = n); v[i] = -(2 - x)*v[i - 2] + x*deriv(v[i - 2], x) + 4*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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA …?
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741