A330510 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 9\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 6 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	6	9							15
2	18	54	36						108
3	54	243	333	135					765
4	162	972	2034	1728	486				5382
5	486	3645	10296	13527	8100	1701			37755
6	1458	13122	46746	83646	78246	35478	5832		264528
7	4374	45927	197649	449010	576153	411642	147987	19683	1852425

All 15 rows

Technical details

Node id: n-442f2d0b70d2a511 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b5f13f55b6382443

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:     2;
  1:     6,      9;
  2:    18,     54,      36;
  3:    54,    243,     333,     135;
  4:   162,    972,    2034,    1728,      486;
  5:   486,   3645,   10296,   13527,     8100,     1701;
  6:  1458,  13122,   46746,   83646,    78246,    35478,     5832;
  7:  4374,  45927,  197649,  449010,   576153,   411642,   147987,   19683;
  8: 13122, 157464,  794700, 2192184,  3590406,  3540024,  2020788,  594864,   65610;
  9: 39366, 531441, 3077928, 9997722, 19966338, 25241139, 19989180, 9408474, 2322594, 216513;

%F Formula

The row polynomials satisfy P(n,x) = -(3+9*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 6 + 9*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 6 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(3 + 9*x + 9*x^2)*P[n - 2, x] + (4 + 6*x)*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] = 2; v[2] = 6 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 9*x + 9*x^2)*v[i - 2] + (4 + 6*x)*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 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzer…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzer…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

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