A035607 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 8\,x\right)\,P_{n-2}(x) + \left(6 + 8\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	11	16													27
3	71	192	128												391
4	481	1888	2432	1024											5825
5	3241	16704	31872	26624	8192										86633
6	21851	139440	352128	439296	270336	65536									1288587
7	147311	1120896	3521280	5840896	5390336	2621440	524288								19166447
8	993121	8775872	32971008	68229120	83935232	61341696	24641536	4194304							285081889
9	6695281	67383168	294606592	730517504	1123123200	1095761920	662175744	226492416	33554432						4240310257
10	45137291	509685584	2541766912	7344871424	13548388352	16537747456	13353091072	6874464256	2046820352	268435456					63070408155
11	304300151	3809689920	21340184448	70412804096	151509057536	222077386752	224497500160	154501382144	69256347648	18253611008	2147483648				938109747511
12	2051487361	28202066976	175304945280	650256792576	1598857691136	2735612624896	3322671529984	2864185344000	1716778958848	681289187328	161061273600	17179869184			13953451771169
13	13830424921	207107151552	1414624649088	5826765813760	16116048207872	31526996672512	44720036773888	46334971215872	34796449300480	18467285630976	6573447446528	1408749273088	137438953472		207543751514009
14	93239986331	1510708542448	11246746369152	50931315600384	156506758553600	344559290351616	559034452672512	676672420446208	610955843076096	406315985797120	193434589593600	62414464745472	12232066859008	1099511627776	3087007394221323

Technical details

Node id: n-27bf97107a5d1d01 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-d98b13d07e12b38e

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:       1;
  2:      11,       16;
  3:      71,      192,       128;
  4:     481,     1888,      2432,      1024;
  5:    3241,    16704,     31872,     26624,       8192;
  6:   21851,   139440,    352128,    439296,     270336,      65536;
  7:  147311,  1120896,   3521280,   5840896,    5390336,    2621440,    524288;
  8:  993121,  8775872,  32971008,  68229120,   83935232,   61341696,  24641536,   4194304;
  9: 6695281, 67383168, 294606592, 730517504, 1123123200, 1095761920, 662175744, 226492416, 33554432;

%F Formula

The row polynomials satisfy P(n,x) = (5+8*x)*P(n-2,x) + (6+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 8*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (5 + 8*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 Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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