A245962 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	2							7
3	7	3							10
4	17	14	2						33
5	31	30	5						66
6	65	89	27	2					183
7	127	210	77	7					421
8	257	542	274	44	2				1119
9	511	1299	792	156	9				2767
10	1025	3182	2430	650	65	2			7354
11	2047	7590	6897	2222	275	11			19042
12	4097	18161	19799	7902	1315	90	2		51366
13	8191	42978	54977	25909	5187	442	13		137697
14	16385	101558	152334	85218	20979	2387	119	2	378982

Technical details

Node id: n-abc4e966333d993f · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d72761e4f44654ed

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,    2;
  3:   7,    3;
  4:  17,   14,   2;
  5:  31,   30,   5;
  6:  65,   89,  27,   2;
  7: 127,  210,  77,   7;
  8: 257,  542, 274,  44, 2;
  9: 511, 1299, 792, 156, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-2,x) + (n-2)*x*d/dx 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 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 2, x] + (n - 2)*x*D[P[n - 2, x], 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 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 2] + (n - 2)*x*deriv(v[i - 2], x) + (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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741