A122542 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	16	16	3													35
4	64	112	52	7												235
5	256	640	512	160	17											1585
6	1024	3328	3840	1984	468	41										10685
7	4096	16384	24832	18304	6992	1328	99									72035
8	16384	77824	146432	141568	76352	23152	3684	239								485635
9	65536	360448	811008	974848	686592	291584	73344	10048	577							3273985
10	262144	1638400	4292608	6184960	5408768	2986496	1045504	224768	27044	1393						22072085
11	1048576	7340032	21954560	36929536	38715392	26484736	12007936	3575040	671248	72016	3363					148802435
12	4194304	32505856	109314048	210436096	257753088	211189760	118355968	45484544	11779648	1963504	190100	8119				1003175035
13	16777216	142606336	532676608	1155530752	1621557248	1552056320	1040457728	493166592	164395776	37673344	5646720	498144	19601			6763062385
14	67108864	620756992	2550136832	6157238272	9748348928	10692788224	8368455680	4738195456	1944210432	572088064	117567232	16009920	1297268	47321		45594249485
15	268435456	2684354560	12029263872	32002539520	56462671872	69909610496	62689181696	41403580416	20266356736	7327522816	1929534208	359434368	44848208	3360368	114243	307380808835

Technical details

Node id: n-c18517aae1169d4a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-97edc89477cdf08e

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 -1)
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:      4,       1;
   3:     16,      16,       3;
   4:     64,     112,      52,       7;
   5:    256,     640,     512,     160,      17;
   6:   1024,    3328,    3840,    1984,     468,      41;
   7:   4096,   16384,   24832,   18304,    6992,    1328,      99;
   8:  16384,   77824,  146432,  141568,   76352,   23152,    3684,    239;
   9:  65536,  360448,  811008,  974848,  686592,  291584,   73344,  10048,   577;
  10: 262144, 1638400, 4292608, 6184960, 5408768, 2986496, 1045504, 224768, 27044, 1393;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(4*x + x^2)*P[n - 2, x] + (4 + 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] = 4 + x; for(n = 3, N, my(i = n); v[i] = (4*x + x^2)*v[i - 2] + (4 + 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 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…?
  - Geometric basis map (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
  - Geometric basis map (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 ?

A122542 → 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