A228955 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	8	6							14
3	48	54	6						108
4	384	600	120						1104
5	3840	8040	2520	120					14520
6	46080	126000	55440	5040					232560
7	645120	2262960	1325520	191520	5040				4430160
8	10321920	45843840	34473600	7015680	362880				98017920
9	185794560	1034570880	974332800	263088000	21772800	362880			2479921920
10	3715891200	25746336000	29817849600	10258617600	1197504000	39916800			70776115200
11	81749606400	700659590400	984148704000	419685235200	64864800000	3472761600	39916800		2254620614400
12	1961990553600	20704844160000	34889997542400	18070814361600	3543174835200	267761894400	6227020800		79444810368000
13	51011754393600	660381782860800	1323497227852800	819781061299200	198069077606400	19789472102400	741015475200	6227020800	3073277618611200
14	1428329123020800	22616053837977600	53526163122432000	39170513910528000	11414253666816000	1442626362777600	76281004800000	1307674368000	129675528702720000

Technical details

Node id: n-214b319cea88310f · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-d79d0ea0a079bfa0

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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,          1;
  2:         8,          6;
  3:        48,         54,         6;
  4:       384,        600,       120;
  5:      3840,       8040,      2520,       120;
  6:     46080,     126000,     55440,      5040;
  7:    645120,    2262960,   1325520,    191520,     5040;
  8:  10321920,   45843840,  34473600,   7015680,   362880;
  9: 185794560, 1034570880, 974332800, 263088000, 21772800, 362880;

%F Formula

The row polynomials satisfy P(n,x) = (n^2-n)*x*P(n-2,x) + (n^2-n)*x*d/dx P(n-2,x) + 2*n*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(n^2 - n)*x*P[n - 2, x] + (n^2 - n)*x*D[P[n - 2, x], x] + 2*n*P[n - 1, x]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = (n^2 - n)*x*v[i - 2] + (n^2 - n)*x*deriv(v[i - 2], x) + 2*n*v[i - 1]); 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 Table: T(n,k) = n!*binomial(n+1,2*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