A081577 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 8\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 4\,x\)

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	4														10
2	46	56	16													118
3	336	608	352	64												1360
4	2476	5920	5152	1920	256											15724
5	18216	54192	62976	35584	9728	1024										181720
6	134056	477024	693504	525824	218624	47104	4096									2100232
7	986496	4086016	7132416	6788608	3796992	1245184	221184	16384								24273280
8	7259536	34304768	69889792	80067584	56329216	24879104	6725632	1015808	65536							280536976
9	53422176	283618880	660570112	884910080	750524416	417419264	152043520	34930688	4587520	262144						3242288800
10	393128416	2316525952	6071232256	9311535104	9246619648	6206038016	2848227328	881721344	176029696	20447232	1048576					37472553568
11	2892992256	18735235584	54568149504	94287801344	107310383104	84401094656	46773305344	18248892416	4908384256	866123776	90177536	4194304				433086734080
12	21289237696	150293669376	481594369536	925684615168	1187971981312	1071681437696	696374788096	328189607936	111259942912	26442989568	4179623936	394264576	16777216			5005373305024
13	156665348736	1197415260928	4186304274432	8859908378624	12657976590336	12884470562816	9607916290048	5311312166912	2175393071104	651626020864	138680467456	19847446528	1711276032	67108864		57849294263680
14	1152884469376	9484403555840	35925779740672	83014268477440	130672689790976	148159319965696	124722579374080	79152432545792	38035723386880	13765837848576	3691927044096	711186186240	92979331072	7381975040	268435456	668589662127232

Technical details

Node id: n-1fa3228c923a928a · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-346585d9f129010f

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,         4;
  2:       46,        56,        16;
  3:      336,       608,       352,        64;
  4:     2476,      5920,      5152,      1920,       256;
  5:    18216,     54192,     62976,     35584,      9728,      1024;
  6:   134056,    477024,    693504,    525824,    218624,     47104,      4096;
  7:   986496,   4086016,   7132416,   6788608,   3796992,   1245184,    221184,    16384;
  8:  7259536,  34304768,  69889792,  80067584,  56329216,  24879104,   6725632,  1015808,   65536;
  9: 53422176, 283618880, 660570112, 884910080, 750524416, 417419264, 152043520, 34930688, 4587520, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (10+8*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[(10 + 8*x)*P[n - 2, x] + (6 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (10 + 8*x)*v[i - 2] + (6 + 4*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)

Related ?

Candidate equalities ?

A081577 → basis A013611 → x ↦ x+1 15 rows; no all-index equality certificate was found

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