A035607 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)

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

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

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	8													19
3	71	96	32												199
4	481	944	608	128											2161
5	3241	8352	7968	3328	512										23401
6	21851	69720	88032	54912	16896	2048									253459
7	147311	560448	880320	730112	336896	81920	8192								2745199
8	993121	4387936	8242752	8528640	5245952	1916928	385024	32768							29733121
9	6695281	33691584	73651648	91314688	70195200	34242560	10346496	1769472	131072						322038001
10	45137291	254842792	635441728	918108928	846774272	516804608	208642048	53706752	7995392	524288					3487978099
11	304300151	1904844960	5335046112	8801600512	9469316096	6939918336	3507773440	1207042048	270532608	35651584	2097152				37778122999
12	2051487361	14101033488	43826236320	81282099072	99928605696	85487894528	51916742656	22376448000	6706167808	1330642944	157286400	8388608			409173032881
13	13830424921	103553575776	353656162272	728345726720	1007253012992	985218646016	698750574592	361991962624	135923630080	36068917248	6419382272	687865856	33554432		4431733435801
14	93239986331	755354271224	2811686592288	6366414450048	9781672409600	10767477823488	8734913323008	5286503284736	2386546262016	793585909760	188900966400	30475812864	2986344448	134217728	47999891653939

Technical details

Node id: n-c247d6abed34f3f9 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-96a3b8d1e1453386

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,        8;
  3:      71,       96,       32;
  4:     481,      944,      608,      128;
  5:    3241,     8352,     7968,     3328,      512;
  6:   21851,    69720,    88032,    54912,    16896,     2048;
  7:  147311,   560448,   880320,   730112,   336896,    81920,     8192;
  8:  993121,  4387936,  8242752,  8528640,  5245952,  1916928,   385024,   32768;
  9: 6695281, 33691584, 73651648, 91314688, 70195200, 34242560, 10346496, 1769472, 131072;

%F Formula

The row polynomials satisfy P(n,x) = (5+4*x)*P(n-2,x) + (6+4*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 + 4*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (5 + 4*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 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)

Related ?

Candidate equalities ?

A035607 → 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