A185263 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	2	1			3
3	6	5			11
4	24	39	8		71
5	120	287	84		491
6	720	2737	1549	180	5186
7	5040	27041	20133	3044	55258

All 15 rows

Technical details

Node id: n-2cfff1a09a0465c5 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-29fb0327734e5ba0

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:      2,       1;
  3:      6,       5;
  4:     24,      39,       8;
  5:    120,     287,      84;
  6:    720,    2737,    1549,     180;
  7:   5040,   27041,   20133,    3044;
  8:  40320,  321545,  363945,  106700,   8064;
  9: 362880, 4018553, 5765001, 2085380, 193248;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (n^3-n^2-n+1+(n^3-n^2)*x)*P(n-2,x) + (n^3-n^2)*x*d/dx P(n-2,x) + (2*n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - n + 1 + (n^3 - n^2)*x)*P[n - 2, x] + (n^3 - n^2)*x*D[P[n - 2, x], x] + (2*n + 1)*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; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 - n + 1 + (n^3 - n^2)*x)*v[i - 2] + (n^3 - n^2)*x*deriv(v[i - 2], x) + (2*n + 1)*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 T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t)…?
  - 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