A245962 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-2}(x) + \left(-2 + n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 5 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	6							11
3	7	9							16
4	17	36	18						71
5	31	75	45						151
6	65	198	189	54					506
7	127	441	504	189					1261
8	257	1032	1476	864	162				3791
9	511	2295	3807	2754	729				10096
10	1025	5130	9855	8910	3645	486			29051
11	2047	11253	24354	25839	13365	2673			79531
12	4097	24588	59454	73224	47385	14580	1458		224786
13	8191	53235	141921	197964	151632	60021	9477		622441
14	16385	114702	334593	522774	466074	231336	56133	4374	1746371

Technical details

Node id: n-d0afb680af3346da · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-9ffd616f7a8b3ed5

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,    6;
  3:   7,    9;
  4:  17,   36,   18;
  5:  31,   75,   45;
  6:  65,  198,  189,   54;
  7: 127,  441,  504,  189;
  8: 257, 1032, 1476,  864, 162;
  9: 511, 2295, 3807, 2754, 729;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(3*n-6)*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 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 5 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (3*n - 6)*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 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (3*n - 6)*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