A081577 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 16\,x\right)\,P_{n-2}(x) + \left(6 + 8\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 6 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	8														14
2	46	112	64													222
3	336	1216	1408	512												3472
4	2476	11840	20608	15360	4096											54380
5	18216	108384	251904	284672	155648	32768										851592
6	134056	954048	2774016	4206592	3497984	1507328	262144									13336168
7	986496	8172032	28529664	54308864	60751872	39845888	14155776	2097152								208847744
8	7259536	68609536	279559168	640540672	901267456	796131328	430440448	130023424	16777216							3270608784
9	53422176	567237760	2642280448	7079280640	12008390656	13357416448	9730785280	4471128064	1174405120	134217728						51218564320
10	393128416	4633051904	24284929024	74492280832	147945914368	198593216512	182286548992	112860332032	45063602176	10468982784	1073741824					802095728864
11	2892992256	37470471168	218272598016	754302410752	1716966129664	2700835028992	2993491542016	2335858229248	1256546369536	443455373312	92341796864	8589934592				12561022876416
12	21289237696	300587338752	1926377478144	7405476921344	19007551700992	34293806006272	44567986438144	42008269815808	28482545385472	13538810658816	4279934910464	807453851648	68719476736			196708809220288
13	156665348736	2394830521856	16745217097728	70879267028992	202527625445376	412303058010112	614906642563072	679847957364736	556900626202624	333632522682368	142008798674944	40647570489344	7009386627072	549755813888		3080509923870848
14	1152884469376	18968807111680	143703118962688	664114147819520	2090763036655616	4741098238902272	7982245079941120	10131511365861376	9737145187041280	7048108978470912	3780533293154304	1456509309419520	380843340070912	60473139527680	4398046511104	48241567973919360

Technical details

Node id: n-270e47a72c949975 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-55137ce63f6b4daa

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:        1;
  1:        6,         8;
  2:       46,       112,         64;
  3:      336,      1216,       1408,        512;
  4:     2476,     11840,      20608,      15360,        4096;
  5:    18216,    108384,     251904,     284672,      155648,       32768;
  6:   134056,    954048,    2774016,    4206592,     3497984,     1507328,     262144;
  7:   986496,   8172032,   28529664,   54308864,    60751872,    39845888,   14155776,    2097152;
  8:  7259536,  68609536,  279559168,  640540672,   901267456,   796131328,  430440448,  130023424,   16777216;
  9: 53422176, 567237760, 2642280448, 7079280640, 12008390656, 13357416448, 9730785280, 4471128064, 1174405120, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (10+16*x)*P(n-2,x) + (6+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[(10 + 16*x)*P[n - 2, x] + (6 + 8*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] = 1; v[2] = 6 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (10 + 16*x)*v[i - 2] + (6 + 8*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 basis_map_A013611 of Pascal-(1,2,1) array read by antidiagonals.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    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 Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (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