A119462 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-8\,x + 4\,n x\right)\,P_{n-2}(x) + \left(-4 + 2\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2\)

\(P_{2}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	8							12
3	8	24							32
4	16	64	32						112
5	32	160	160						352
6	64	384	576	128					1152
7	128	896	1792	896					3712
8	256	2048	5120	4096	512				12032
9	512	4608	13824	15360	4608				38912
10	1024	10240	35840	51200	25600	2048			125952
11	2048	22528	90112	157696	112640	22528			407552
12	4096	49152	221184	458752	430080	147456	8192		1318912
13	8192	106496	532480	1277952	1490944	745472	106496		4268032
14	16384	229376	1261568	3440640	4816896	3211264	802816	32768	13811712

Technical details

Node id: n-08e8485c39053ba9 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-088313a0be276231

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   2;
  2:   4,    8;
  3:   8,   24;
  4:  16,   64,    32;
  5:  32,  160,   160;
  6:  64,  384,   576,   128;
  7: 128,  896,  1792,   896;
  8: 256, 2048,  5120,  4096,  512;
  9: 512, 4608, 13824, 15360, 4608;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (4*n-8)*x*P(n-2,x) + (2*n-4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8)*x*P[n - 2, x] + (2*n - 4)*P[n - 1, x])/(n - 2)]];
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] = 2; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8)*x*v[i - 2] + (2*n - 4)*v[i - 1])/(n - 2)); 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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