A146986 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(24 + 72\,x + 54\,x^{2} - 8\,n - 24\,n x - 18\,n x^{2}\right)\,P_{n-2}(x) + \left(-18 - 27\,x + 6\,n + 9\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 6 + 18\,x + 9\,x^{2}\)

\(P_{3}(x) = 16 + 72\,x + 90\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	3							5
2	6	18	9						33
3	16	72	90	27					205
4	48	288	576	432	81				1425
5	160	1200	3312	4104	2106	243			11125
6	576	5184	18288	31968	28188	10692	729		95625
7	2176	22848	98208	222480	281880	191484	56862	2187	878125

All 15 rows

Technical details

Node id: n-0bec1a9f266d1c7a · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-25152b6183118bbf

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:     2,      3;
  2:     6,     18,       9;
  3:    16,     72,      90,      27;
  4:    48,    288,     576,     432,       81;
  5:   160,   1200,    3312,    4104,     2106,      243;
  6:   576,   5184,   18288,   31968,    28188,    10692,      729;
  7:  2176,  22848,   98208,  222480,   281880,   191484,    56862,    2187;
  8:  8448, 101376,  513792, 1430784,  2371680,  2348352,  1294704,  314928,    6561;
  9: 33280, 449280, 2621952, 8660736, 17744832, 23133600, 18779040, 8713008, 1797714, 19683;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(8*n-24+(24*n-72)*x+(18*n-54)*x^2)*P(n-2,x) + (6*n-18+(9*n-27)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 6 + 18*x + 9*x^2, P(3,x) = 16 + 72*x + 90*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 6 + 18*x + 9*x^2;
P[3, x_] := 16 + 72*x + 90*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (24*n - 72)*x + (18*n - 54)*x^2)*P[n - 2, x] + (6*n - 18 + (9*n - 27)*x)*P[n - 1, x])/(n - 3)]];
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] = 2 + 3*x; v[3] = 6 + 18*x + 9*x^2; v[4] = 16 + 72*x + 90*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 24 + (24*n - 72)*x + (18*n - 54)*x^2)*v[i - 2] + (6*n - 18 + (9*n - 27)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.?
  - 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