A081579 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(12 + 16\,x\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	4														8
2	28	48	16													92
3	160	416	320	64												960
4	976	3328	3904	1792	256											10256
5	5824	24768	39424	28672	9216	1024										108928
6	35008	177920	356864	356352	183296	45056	4096									1158592
7	209920	1242112	3008512	3827712	2727936	1073152	212992	16384								12318720
8	1259776	8503296	24131584	37330944	34123776	18677760	5914624	983040	65536							130990336
9	7558144	57316352	186515456	339918848	379797504	267730944	118095872	31195136	4456448	262144						1392846848
10	45349888	381693952	1400958976	2939813888	3885645824	3360227328	1913126912	703594496	159121408	19922944	1048576					14810504192
11	272097280	2516901888	10285858816	24426364928	37298110464	38273024000	26794262528	12730761216	4003463168	790626304	88080384	4194304				157483745280
12	1632587776	16461922304	74129653760	196542005248	340562673664	404777598976	336990306304	197153259520	80103866368	21961375744	3846176768	385875968	16777216			1674564079616
13	9795518464	106934419456	526067040256	1540376756224	2986817880064	4037407145984	3900971155456	2718051598336	1360762241024	481803894784	116937195520	18387828736	1677721600	67108864		17806057504768
14	58773127168	690584223744	3684952440832	11810353709056	25340202713088	38403234070528	42273838465024	34233775030272	20430953906176	8915462914048	2792500494336	607469436928	86637543424	7247757312	268435456	189336254267392

Technical details

Node id: n-2e4f18459aae0e07 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-7d9acbbdb3cf6a7d

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:       4,        4;
  2:      28,       48,        16;
  3:     160,      416,       320,        64;
  4:     976,     3328,      3904,      1792,       256;
  5:    5824,    24768,     39424,     28672,      9216,      1024;
  6:   35008,   177920,    356864,    356352,    183296,     45056,      4096;
  7:  209920,  1242112,   3008512,   3827712,   2727936,   1073152,    212992,    16384;
  8: 1259776,  8503296,  24131584,  37330944,  34123776,  18677760,   5914624,   983040,   65536;
  9: 7558144, 57316352, 186515456, 339918848, 379797504, 267730944, 118095872, 31195136, 4456448, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (12+16*x)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(12 + 16*x)*P[n - 2, x] + (4 + 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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (12 + 16*x)*v[i - 2] + (4 + 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_A013609 of Pascal-(1,4,1) array.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,4,1) array.?
    
    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-(1,4,1) array.?
    
    ↑ 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-(1,4,1) array.?
    
    ↑ 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-(1,4,1) array.?
    
    ↑ 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