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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	149505	572800	946176	879072	501792	179968	39552	4864	256							3273985
10	694563	2980114	5609920	6075520	4167648	1876032	553600	103168	11008	512						22072085
11	3226767	15326992	32384036	40051488	32085152	17382912	6445056	1613568	260864	24576	1024					148802435
12	14990757	78090970	182980116	254420424	233097568	147665344	65958912	20761088	4509952	643584	54272	2048				1003175035
13	69643329	394779904	1015908496	1568564464	1617589552	1173175552	613267328	232692992	63571712	12189696	1556480	118784	4096			6763062385
14	323545587	1982660698	5558316080	9435296352	10815620688	8837072480	5292627840	2351507712	774724864	186852864	32055296	3702784	258048	8192		45594249485
15	1503112335	9901360336	30035871012	55595327952	70122973040	63734237056	43030809408	21915900160	8458016000	2460573696	530975744	82391040	8687616	557056	16384	307380808835

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