A266213 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)

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

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

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	9	6													15
3	49	60	18												127
4	281	498	288	54											1121
5	1601	3720	3186	1188	162										9857
6	9129	26238	29736	16578	4536	486									86703
7	52049	178260	251298	186408	76626	16524	1458								762623
8	296761	1179786	1988928	1841454	1010232	328050	58320	4374							6707905
9	1692001	7658400	15023970	16673580	11441250	4966920	1331154	201204	13122						59001601
10	9647049	48977430	109590120	141772410	116792280	63501246	22773960	5191938	682344	39366					518968143
11	55003249	309537900	777953970	1149398640	1105064370	722074500	324598914	99079848	19643634	2283228	118098				4564756351
12	313604441	1937550114	5403676320	8976715110	9866004120	7529947434	4070817648	1558285614	413762904	72551538	7558272	354294			40150827809
13	1788035201	12031725240	36871461522	68026060980	84128619330	73451240640	46408549698	21373997148	7119485442	1672110216	262768050	24800580	1062882		353159916929
14	10194593769	74213745582	247879838952	502862578866	691115441400	679359863286	491269583304	264541230234	106049327112	31250502270	6578058600	936399042	80779032	3188646	3106335130095

Technical details

Node id: n-d7889f3e929cbf0f · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-5d09cadc540fdad8

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:       9,       6;
  3:      49,      60,       18;
  4:     281,     498,      288,       54;
  5:    1601,    3720,     3186,     1188,      162;
  6:    9129,   26238,    29736,    16578,     4536,     486;
  7:   52049,  178260,   251298,   186408,    76626,   16524,    1458;
  8:  296761, 1179786,  1988928,  1841454,  1010232,  328050,   58320,   4374;
  9: 1692001, 7658400, 15023970, 16673580, 11441250, 4966920, 1331154, 201204, 13122;

%F Formula

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-2,x) + (5+3*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[(4 + 3*x)*P[n - 2, x] + (5 + 3*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] = (4 + 3*x)*v[i - 2] + (5 + 3*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_A013610 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 ?

A266213 → basis A013610 → 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