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	x⁵	x⁶	x⁷	x⁸	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
8	512	6372	30048	65280	61440	16384				180036
9	1024	14108	75840	197376	245760	114688				648796
10	2048	30948	186912	568320	890880	638976	131072			2449156
11	4096	67356	451456	1569792	2961408	2834432	1048576			8937116
12	8192	145636	1072864	4201728	9289728	11239424	6422528	1048576		33428676
13	16384	313116	2514624	10952960	27770880	40599552	31195136	9437184		122799836
14	32768	669924	5825440	27939328	79964160	137396224	133955584	62914560	8388608	457086596

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