A245962 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 5 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	8							13
3	7	12							19
4	17	48	32						97
5	31	100	80						211
6	65	264	336	128					793
7	127	588	896	448					2059
8	257	1376	2624	2048	512				6817
9	511	3060	6768	6528	2304				19171
10	1025	6840	17520	21120	11520	2048			60073
11	2047	15004	43296	61248	42240	11264			175099
12	4097	32784	105696	173568	149760	61440	8192		535537
13	8191	70980	252304	469248	479232	252928	53248		1586131
14	16385	152936	594832	1239168	1473024	974848	315392	32768	4799353

Technical details

Node id: n-c0fed11cd1042315 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e249835bd0d780d4

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:   1;
  2:   5,    8;
  3:   7,   12;
  4:  17,   48,   32;
  5:  31,  100,   80;
  6:  65,  264,  336,  128;
  7: 127,  588,  896,  448;
  8: 257, 1376, 2624, 2048,  512;
  9: 511, 3060, 6768, 6528, 2304;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(4*n-8)*x)*P(n-2,x) + (n-2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 5 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (4*n - 8)*x)*P[n - 2, x] + (n - 2)*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] = 1; v[3] = 5 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (4*n - 8)*x)*v[i - 2] + (n - 2)*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 induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the…?
  - 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