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	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

All 15 rows

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