A122542 → reverse → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1	1														1
2	1														1
3	3	8													11
4	7	32	32												71
5	17	112	224	128											481
6	41	352	1056	1280	512										3241
7	99	1048	4192	7808	6656	2048									21851
8	239	3008	15040	37888	50176	32768	8192								147311
9	577	8416	50496	160512	289792	294912	155648	32768							993121
10	1393	23104	161728	621056	1423360	1982464	1630208	720896	131072						6695281
11	3363	62504	500032	2251520	6262784	11112448	12525568	8617984	3276800	524288					45137291
12	8119	167136	1504224	7771136	25439232	54951936	79060992	74579968	44040192	14680064	2097152				304300151
13	19601	442704	4427040	25810816	97231872	247824384	434905088	524124160	424148992	219152384	65011712	8388608			2051487361
14	47321	1163424	12797664	83117824	354230784	1041285120	2159976448	3178692608	3287154688	2325741568	1067450368	285212672	33554432		13830424921
15	114243	3037240	36446880	260945280	1241408000	4136245248	9911296000	17261035520	21809725440	19715850240	12379488256	5108662272	1241513984	134217728	93239986331

Technical details

Node id: n-28b4b73ea763f8fe · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-97f0a8adbf46de41

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    1;
   2:    1;
   3:    3,     8;
   4:    7,    32,     32;
   5:   17,   112,    224,    128;
   6:   41,   352,   1056,   1280,     512;
   7:   99,  1048,   4192,   7808,    6656,    2048;
   8:  239,  3008,  15040,  37888,   50176,   32768,    8192;
   9:  577,  8416,  50496, 160512,  289792,  294912,  155648,  32768;
  10: 1393, 23104, 161728, 621056, 1423360, 1982464, 1630208, 720896, 131072;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 2, x] + (2 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (1 + 4*x)*v[i - 2] + (2 + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A035607 → basis A013611

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