A081577 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-2}(x) + \left(5 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	3														8
2	33	36	9													78
3	205	333	171	27												736
4	1289	2766	2142	702	81											6980
5	8085	21591	22374	11178	2673	243										66144
6	50737	162072	210375	141480	51759	9720	729									626872
7	318365	1183809	1846629	1562193	771687	221859	34263	2187								5940992
8	1997721	8475138	15440004	15744942	9807966	3812670	901044	118098	6561							56304144
9	12535525	59749515	124501320	148622040	111821310	54892242	17548488	3516696	400221	19683						533607040
10	78659393	416141580	976126005	1335213720	1177906050	699274296	282503538	76579992	13312269	1338444	59049					5057114336
11	493581165	2869895349	7483562415	11540430855	11681474130	8140155426	3980081934	1363835070	320603265	49187817	4428675	177147				47927413248
12	3097180969	19631309238	56333355642	96691307310	110463193935	88406830188	50776550028	21067082316	6260499639	1298330046	178209882	14526054	531441			454218906688
13	19434554165	133368738879	417648577338	789681424806	1005084269775	908633820933	599784828732	292456233828	105251581683	27590271273	5116595850	635249142	47298249	1594323		4304735038976
14	121950218577	900780916608	3056803803891	6312883448376	8858319018633	8928455719104	6661478987811	3734831614032	1580113100907	501655739616	117569452401	19711500984	2233646523	153055008	4782969	40796945005440

Technical details

Node id: n-c0bd052e344cc31b · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-9f5d00811463c524

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:        5,        3;
  2:       33,       36,         9;
  3:      205,      333,       171,        27;
  4:     1289,     2766,      2142,       702,        81;
  5:     8085,    21591,     22374,     11178,      2673,      243;
  6:    50737,   162072,    210375,    141480,     51759,     9720,      729;
  7:   318365,  1183809,   1846629,   1562193,    771687,   221859,    34263,    2187;
  8:  1997721,  8475138,  15440004,  15744942,   9807966,  3812670,   901044,  118098,   6561;
  9: 12535525, 59749515, 124501320, 148622040, 111821310, 54892242, 17548488, 3516696, 400221, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (8+6*x)*P(n-2,x) + (5+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 6*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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 6*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 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 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