A305540 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 5\,x - 3\,x^{2} + 2\,x^{3}\right)\,P_{n-4}(x) + \left(-x - 2\,x^{2} + x^{4}\right)\,P_{n-4}'(x) + \left(3 + 5\,x - x^{2}\right)\,P_{n-2}(x) + \left(x + x^{2}\right)\,P_{n-2}'(x)\)

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

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

\(P_{2}(x) = 3 + 2\,x\)

\(P_{3}(x) = 5 + 10\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	3	2							5
3	5	10	3						18
4	7	18	6						31
5	11	52	57	12					132
6	15	86	108	24					233
7	23	208	489	372	60				1152
8	31	330	870	720	120				2071
9	47	736	3045	4740	2760	360			11688
10	63	1142	5220	8760	5400	720			21305
11	95	2440	16113	42180	48900	23040	2520		135288
12	127	3738	27006	75600	92400	45360	5040		249271
13	191	7792	77637	310212	588000	541800	214200	20160	1759992
14	255	11846	128268	544824	1083600	1038240	423360	40320	3270713

Technical details

Node id: n-79838d1a829232ee · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e5e6ab1c9af2894f

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:  3,   2;
  3:  5,  10,    3;
  4:  7,  18,    6;
  5: 11,  52,   57,   12;
  6: 15,  86,  108,   24;
  7: 23, 208,  489,  372,   60;
  8: 31, 330,  870,  720,  120;
  9: 47, 736, 3045, 4740, 2760, 360;

%F Formula

The row polynomials satisfy P(n,x) = -(2+5*x+3*x^2-2*x^3)*P(n-4,x) - (x+2*x^2-x^4)*d/dx P(n-4,x) + (3+5*x-x^2)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 3 + 2*x, P(3,x) = 5 + 10*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 3 + 2*x;
P[3, x_] := 5 + 10*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(2 + 5*x + 3*x^2 - 2*x^3)*P[n - 4, x] - (x + 2*x^2 - x^4)*D[P[n - 4, x], x] + (3 + 5*x - x^2)*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] = 1; v[2] = 2 + x; v[3] = 3 + 2*x; v[4] = 5 + 10*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(2 + 5*x + 3*x^2 - 2*x^3)*v[i - 4] - (x + 2*x^2 - x^4)*deriv(v[i - 4], x) + (3 + 5*x - x^2)*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.