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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	13122	157464	794700	2192184	3590406	3540024	2020788	594864	65610							12969162
9	39366	531441	3077928	9997722	19966338	25241139	19989180	9408474	2322594	216513						90790695
10	118098	1771470	11580984	43312428	102198510	158099202	161167320	105737076	42003522	8857350	708588					635554548
11	354294	5845851	42581709	180258129	491088744	900187911	1126429659	954653931	530889876	181221381	33126489	2302911				4448940885
12	1062882	19131876	153652878	726413400	2245267512	4763388060	7080664734	7386583752	5323332960	2553987348	759960630	121877136	7440174			31142763342
13	3188646	62178597	545856336	2850948441	9860725674	23782471722	41002208424	50926803489	45290412414	28242388746	11855385828	3111232761	442158912	23914845		217999874835
14	9565938	200884698	1913785470	10944628794	41892194304	113258948544	222383769678	320964238098	340547235336	262660817016	143700111126	53386082802	12477171798	1584757062	76527504	1526000718168

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