A199478 → basis A013611

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 2 + 4\,x\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	2	4				6
1	4	12				16
2	8	36	32			76
3	16	92	128			236
4	32	228	480	256		996
5	64	540	1472	1280		3356
6	128	1252	4256	5632	2048	13316
7	256	2844	11520	19968	12288	46876

All 15 rows

Technical details

Node id: n-c8021aa0b636ccac · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-61e4d58d94cdaa75

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    2,     4;
  1:    4,    12;
  2:    8,    36,    32;
  3:   16,    92,   128;
  4:   32,   228,   480,    256;
  5:   64,   540,  1472,   1280;
  6:  128,  1252,  4256,   5632,   2048;
  7:  256,  2844, 11520,  19968,  12288;
  8:  512,  6372, 30048,  65280,  61440,  16384;
  9: 1024, 14108, 75840, 197376, 245760, 114688;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + 4*x;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 2, 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] = 2 + 4*x; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 2] + 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 Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.?
  - 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