A308737 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-8 - 32\,x - 32\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 12\,x\)

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	12														16
2	16	88	112													216
3	64	496	1216	960												2736
4	256	2528	9024	13952	7936											33696
5	1024	12224	56832	129536	145408	64512										409536
6	4096	57216	326400	977920	1628160	1431552	520192									4945536
7	16384	261888	1766400	6538240	14376960	18812928	13582336	4177920								59533056
8	65536	1179136	9167872	40327168	109957120	190554112	205176832	125599744	33488896							715516416
9	262144	5241856	46120960	234766336	762904576	1643020288	2346975232	2145648640	1139802112	268173312						8592915456
10	1048576	23066624	226455552	1308327936	4931125248	12679446528	22540320768	27369406464	21733834752	10195828736	2146435072					103155296256
11	4194304	100659200	1090437120	7045693440	30195056640	90180550656	191629885440	289855242240	305953505280	214701178880	90173341696	17175674880				1238105419776
12	16777216	436199424	5167202304	36908072960	177156587520	602325909504	1488085057536	2692702273536	3543001989120	3306778787840	2078533484544	790181707776	137422176256			14858716225536
13	67108864	1879031808	24158797824	188974235648	1003919769600	3826686099456	10771362742272	22676458438656	35715287089152	41573207244800	34770209734656	19790102003712	6871545020416	1099444518912		178313301835776
14	268435456	8053030912	111668297728	949177548800	5527547936768	23338477420544	73700289282048	176880334405632	324280013291520	452025921830912	471678492606464	357331682459648	185812230602752	59371883069440	8795824586752	2139811864805376

Technical details

Node id: n-dfff3db745a4b454 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-762c02f66233cc77

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,      12;
  2:     16,      88,      112;
  3:     64,     496,     1216,       960;
  4:    256,    2528,     9024,     13952,      7936;
  5:   1024,   12224,    56832,    129536,    145408,      64512;
  6:   4096,   57216,   326400,    977920,   1628160,    1431552,     520192;
  7:  16384,  261888,  1766400,   6538240,  14376960,   18812928,   13582336,    4177920;
  8:  65536, 1179136,  9167872,  40327168, 109957120,  190554112,  205176832,  125599744,   33488896;
  9: 262144, 5241856, 46120960, 234766336, 762904576, 1643020288, 2346975232, 2145648640, 1139802112, 268173312;

%F Formula

The row polynomials satisfy P(n,x) = -(8+32*x+32*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 32*x + 32*x^2)*P[n - 2, x] + (6 + 12*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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 32*x + 32*x^2)*v[i - 2] + (6 + 12*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