A122542 → reverse → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

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	4													7
4	7	16	8												31
5	17	56	56	16											145
6	41	176	264	160	32										673
7	99	524	1048	976	416	64									3127
8	239	1504	3760	4736	3136	1024	128								14527
9	577	4208	12624	20064	18112	9216	2432	256							67489
10	1393	11552	40432	77632	88960	61952	25472	5632	512						313537
11	3363	31252	125008	281440	391424	347264	195712	67328	12800	1024					1456615
12	8119	83568	376056	971392	1589952	1717248	1235328	582656	172032	28672	2048				6767071
13	19601	221352	1106760	3226352	6076992	7744512	6795392	4094720	1656832	428032	63488	4096			31438129
14	47321	581712	3199416	10389728	22139424	32540160	33749632	24833536	12840448	4542464	1042432	139264	8192		146053729
15	114243	1518620	9111720	32618160	77588000	129257664	154864000	134851840	85194240	38507520	12089344	2494464	303104	16384	678529303

Technical details

Node id: n-35a0c1ad8f0f3856 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9740049352638c67

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,     4;
   4:    7,    16,     8;
   5:   17,    56,    56,    16;
   6:   41,   176,   264,   160,    32;
   7:   99,   524,  1048,   976,   416,    64;
   8:  239,  1504,  3760,  4736,  3136,  1024,   128;
   9:  577,  4208, 12624, 20064, 18112,  9216,  2432,  256;
  10: 1393, 11552, 40432, 77632, 88960, 61952, 25472, 5632, 512;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (2+2*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 + 2*x)*P[n - 2, x] + (2 + 2*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 + 2*x)*v[i - 2] + (2 + 2*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 A013609

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