A199478 → basis A013610

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 2 + 3\,x\)

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	2	3				5
1	4	9				13
2	8	27	18			53
3	16	69	72			157
4	32	171	270	108		581
5	64	405	828	540		1837
6	128	939	2394	2376	648	6485
7	256	2133	6480	8424	3888	21181

All 15 rows

Technical details

Node id: n-efb3c12c83e8f5e7 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fefee1d2617f2700

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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,     3;
  1:    4,     9;
  2:    8,    27,    18;
  3:   16,    69,    72;
  4:   32,   171,   270,   108;
  5:   64,   405,   828,   540;
  6:  128,   939,  2394,  2376,   648;
  7:  256,  2133,  6480,  8424,  3888;
  8:  512,  4779, 16902, 27540, 19440,  3888;
  9: 1024, 10581, 42660, 83268, 77760, 27216;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + 3*x;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*x)*P[n - 2, 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 + 3*x; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*x)*v[i - 2] + 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 Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/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