A121574 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	40	39	9													88
3	264	384	180	27												855
4	1744	3372	2385	729	81											8311
5	11520	27792	26298	12177	2754	243										80784
6	76096	220032	260820	162027	55566	9963	729									785233
7	502656	1694208	2413584	1882224	867024	235710	34992	2187								7632585
8	3320320	12781632	21267504	19964664	11557161	4221882	949887	120285	6561							74189896
9	21932544	94935552	180686880	198360144	138351726	63546687	19212066	3685095	406782	19683						721137159
10	144876544	696498432	1492342848	1875882672	1531313424	847894311	322377651	83077569	13883076	1358127	59049					7009563703
11	956989440	5059160064	12051106560	17067825792	15964015896	10350548064	4745437164	1537974945	345213576	51097068	4487724	177147				68134033440
12	6321442816	36446552064	95542986240	150540833664	158740474464	117980853588	63357460713	24763604391	6989971302	1389304872	184528125	14703201	531441			662273246881
13	41756614656	260751249408	745979480064	1294298583552	1519124888736	1273400784864	784120717890	359042219757	122225420124	30545772138	5446325466	655975341	47829690	1594323		6437397456009
14	275825459200	1854527877120	5749642229760	10892522234880	14084229481920	13145924213136	9132299065548	4795742271915	1912729878396	576478026594	129221296326	20887225623	2301139530	154649331	4782969	62572489832248

Technical details

Node id: n-483031b65a8f51fe · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-ad215bfd092541fc

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:        6,        3;
  2:       40,       39,         9;
  3:      264,      384,       180,        27;
  4:     1744,     3372,      2385,       729,        81;
  5:    11520,    27792,     26298,     12177,      2754,      243;
  6:    76096,   220032,    260820,    162027,     55566,     9963,      729;
  7:   502656,  1694208,   2413584,   1882224,    867024,   235710,    34992,    2187;
  8:  3320320, 12781632,  21267504,  19964664,  11557161,  4221882,   949887,  120285,   6561;
  9: 21932544, 94935552, 180686880, 198360144, 138351726, 63546687, 19212066, 3685095, 406782, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[(4 + 3*x)*P[n - 2, x] + (6 + 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] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 3*x)*v[i - 2] + (6 + 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 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ 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 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Candidate equalities ?

A121574 → 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