A206306 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 4 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	4	5	1						10
3	14	22	9	1					46
4	48	92	56	13	1				210
5	164	372	298	106	17	1			958
6	560	1468	1452	696	172	21	1		4370
7	1912	5688	6680	4024	1350	254	25	1	19934

All 15 rows

Technical details

Node id: n-f6a21c7b563b30ff · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-163559e669a88e09

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,     1;
  2:     4,     5,      1;
  3:    14,    22,      9,      1;
  4:    48,    92,     56,     13,     1;
  5:   164,   372,    298,    106,    17,     1;
  6:   560,  1468,   1452,    696,   172,    21,    1;
  7:  1912,  5688,   6680,   4024,  1350,   254,   25,   1;
  8:  6528, 21728,  29504,  21384,  9080,  2324,  352,  29,  1;
  9: 22288, 82064, 126384, 106992, 55004, 17868, 3682, 466, 33, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4)*P(n-2,x) + (4*n-8+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 4 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4)*P[n - 2, x] + (4*n - 8 + (n - 2)*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 + x; v[3] = 4 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4)*v[i - 2] + (4*n - 8 + (n - 2)*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