A305540 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 6\,x - 6\,x^{2}\right)\,P_{n-4}(x) + \left(-x - 3\,x^{2} - 2\,x^{3}\right)\,P_{n-4}'(x) + \left(3 + 6\,x\right)\,P_{n-2}(x) + \left(x + x^{2}\right)\,P_{n-2}'(x)\)

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

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

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

\(P_{3}(x) = 4 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0					0
1	1				1
2	2				2
3	4	6			10
4	6	12			18
5	10	42	36		88
6	14	72	72		158
7	22	186	396	240	844

All 15 rows

Technical details

Node id: n-7ee6c371ec3eafc5 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-94258c32f1961a83

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (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:  0;
  1:  1;
  2:  2;
  3:  4,   6;
  4:  6,  12;
  5: 10,  42,   36;
  6: 14,  72,   72;
  7: 22, 186,  396,  240;
  8: 30, 300,  720,  480;
  9: 46, 690, 2700, 3840, 1800;

%F Formula

The row polynomials satisfy P(n,x) = -(2+6*x+6*x^2)*P(n-4,x) - (x+3*x^2+2*x^3)*d/dx P(n-4,x) + (3+6*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2, P(3,x) = 4 + 6*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 2;
P[3, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 6*x + 6*x^2)*P[n - 4, x] - (x + 3*x^2 + 2*x^3)*D[P[n - 4, x], x] + (3 + 6*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 2; v[4] = 4 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = -(2 + 6*x + 6*x^2)*v[i - 4] - (x + 3*x^2 + 2*x^3)*deriv(v[i - 4], x) + (3 + 6*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k diffe…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.