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	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

All 15 rows

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