A152830 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

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

\(P_{3}(x) = 1\)

Rows ?

n	x⁰	x¹	P_n(1)
0	1		1
1	1		1
2	1		1
3	1		1
4	4	2	6
5	4	2	6
6	4	2	6
7	4	2	6

All 15 rows

Technical details

Node id: n-993b9927637e1833 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-58603df563d92c86

Observed on the stored rows ?

Degrees: 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3 (irregular)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  1;
  2:  1;
  3:  1;
  4:  4,  2;
  5:  4,  2;
  6:  4,  2;
  7:  4,  2;
  8: 16, 16, 4;
  9: 16, 16, 4;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 2*x)*P[n - 4, 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] = 1; v[3] = 1; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (4 + 2*x)*v[i - 4]); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, A007318 rows repeated four times .?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, A007318 rows repeated four times .?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, A007318 rows repeated four times .?
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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.