A122542 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	15	16	4						35
4	69	106	52	8					235
5	321	640	464	144	16				1585
6	1491	3658	3504	1632	368	32			10685
7	6927	20176	24004	14896	5072	896	64		72035
8	32181	108514	154196	119496	54448	14560	2112	128	485635

All 15 rows

Technical details

Node id: n-ac703ca87f07f4be · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-af8be86364e13872

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      3,       2;
   3:     15,      16,       4;
   4:     69,     106,      52,       8;
   5:    321,     640,     464,     144,      16;
   6:   1491,    3658,    3504,    1632,     368,      32;
   7:   6927,   20176,   24004,   14896,    5072,     896,     64;
   8:  32181,  108514,  154196,  119496,   54448,   14560,   2112,    128;
   9: 149505,  572800,  946176,  879072,  501792,  179968,  39552,   4864,   256;
  10: 694563, 2980114, 5609920, 6075520, 4167648, 1876032, 553600, 103168, 11008, 512;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (3 + 2*x)*v[i - 2] + (4 + 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by 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 basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - ↑ Geometric basis map (elementary) (above)
  - ↑ Geometric basis map (elementary) (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