A308737 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-8 - 16\,x - 8\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	6														10
2	16	44	28													88
3	64	248	304	120												736
4	256	1264	2256	1744	496											6016
5	1024	6112	14208	16192	9088	2016										48640
6	4096	28608	81600	122240	101760	44736	8128									391168
7	16384	130944	441600	817280	898560	587904	212224	32640								3137536
8	65536	589568	2291968	5040896	6872320	5954816	3205888	981248	130816							25133056
9	262144	2620928	11530240	29345792	47681536	51344384	36671488	16762880	4452352	523776						201195520
10	1048576	11533312	56613888	163540992	308195328	396232704	352192512	213823488	84897792	19913728	2096128					1610088448
11	4194304	50329600	272609280	880711680	1887191040	2818142208	2994216960	2264494080	1195130880	419338240	88059904	8386560				12882804736
12	16777216	218099712	1291800576	4613509120	11072286720	18822684672	23251329024	21036736512	13839851520	6458552320	2029817856	385830912	33550336			103070826496
13	67108864	939515904	6039699456	23621779456	62744985600	119583940608	168302542848	177159831552	139512840192	81197670400	33955282944	9663135744	1677623296	134209536		824600166400
14	268435456	4026515456	27917074432	118647193600	345471746048	729327419392	1151567020032	1381877612544	1266718801920	882863128576	460623527936	174478360576	45364314112	7247544320	536854528	6596935548928

Technical details

Node id: n-d6a51c5da97a7289 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-19181af399122da8

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      4,       6;
  2:     16,      44,       28;
  3:     64,     248,      304,      120;
  4:    256,    1264,     2256,     1744,      496;
  5:   1024,    6112,    14208,    16192,     9088,     2016;
  6:   4096,   28608,    81600,   122240,   101760,    44736,     8128;
  7:  16384,  130944,   441600,   817280,   898560,   587904,   212224,    32640;
  8:  65536,  589568,  2291968,  5040896,  6872320,  5954816,  3205888,   981248,  130816;
  9: 262144, 2620928, 11530240, 29345792, 47681536, 51344384, 36671488, 16762880, 4452352, 523776;

%F Formula

The row polynomials satisfy P(n,x) = -(8+16*x+8*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 16*x + 8*x^2)*P[n - 2, x] + (6 + 6*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] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 16*x + 8*x^2)*v[i - 2] + (6 + 6*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 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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