A119462 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	2							6
3	8	6							14
4	16	18	2						36
5	32	50	10						92
6	64	134	42	2					242
7	128	350	154	14					646
8	256	898	526	76	2				1758
9	512	2274	1710	348	18				4862
10	1024	5702	5370	1450	120	2			13668
11	2048	14190	16434	5654	660	22			39008
12	4096	35106	49310	21028	3250	174	2		112966
13	8192	86450	145678	75452	14794	1118	26		331710
14	16384	212102	425082	263298	63616	6356	238	2	987078

Technical details

Node id: n-4bc40ad339c7a125 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d79d0ea0a079bfa0

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,    2;
  3:   8,    6;
  4:  16,   18,    2;
  5:  32,   50,   10;
  6:  64,  134,   42,   2;
  7: 128,  350,  154,  14;
  8: 256,  898,  526,  76,  2;
  9: 512, 2274, 1710, 348, 18;

%F Formula

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

%t Mathematica

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