A208509 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	6	1							7
3	12	5							17
4	24	17	1						42
5	48	51	7						106
6	96	143	33	1					273
7	192	385	131	9					717
8	384	1009	471	54	1				1919
9	768	2595	1589	266	11				5229
10	1536	6583	5129	1165	80	1			14494
11	3072	16529	16031	4717	470	13			40832
12	6144	41177	48903	18058	2425	111	1		116819
13	12288	101955	146397	66298	11447	757	15		339157
14	24576	251231	431777	235673	50652	4494	147	1	998551

Technical details

Node id: n-2cb7ff9fe3f3bf65 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-23551e2957101112

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 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:   3;
  2:   6,    1;
  3:  12,    5;
  4:  24,   17,    1;
  5:  48,   51,    7;
  6:  96,  143,   33,   1;
  7: 192,  385,  131,   9;
  8: 384, 1009,  471,  54,  1;
  9: 768, 2595, 1589, 266, 11;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[x*P[n - 2, x] + x*D[P[n - 2, x], x] + 2*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 2] + x*deriv(v[i - 2], x) + 2*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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.?
  - 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