A128908 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 3 + 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	3	4	1						8
3	8	14	7	1					30
4	21	46	34	10	1				112
5	55	145	141	63	13	1			418
6	144	444	534	320	101	16	1		1560
7	377	1331	1905	1431	610	148	19	1	5822

All 15 rows

Technical details

Node id: n-3b0e9ba368b8ed81 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-00c723c1b86b42df

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:    3,     4,     1;
  3:    8,    14,     7,     1;
  4:   21,    46,    34,    10,     1;
  5:   55,   145,   141,    63,    13,    1;
  6:  144,   444,   534,   320,   101,   16,    1;
  7:  377,  1331,  1905,  1431,   610,  148,   19,   1;
  8:  987,  3926,  6512,  5878,  3160, 1038,  204,  22,  1;
  9: 2584, 11434, 21557, 22715, 14748, 6126, 1631, 269, 25, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2)*P(n-2,x) + (3*n-6+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*P[n - 2, x] + (3*n - 6 + (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] = 3 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2)*v[i - 2] + (3*n - 6 + (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-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