A133081 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	0	2	3					5
3	0	0	3					3
4	0	0	0	4	5			9
5	0	0	0	0	5			5
6	0	0	0	0	0	6	7	13
7	0	0	0	0	0	0	7	7

All 15 rows

Technical details

Node id: n-54e15ef14165387b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1d293b5070e22c3c

Observed on the stored rows ?

Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (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: 0, 2, 3;
  3: 0, 0, 3;
  4: 0, 0, 0, 4, 5;
  5: 0, 0, 0, 0, 5;
  6: 0, 0, 0, 0, 0, 6, 7;
  7: 0, 0, 0, 0, 0, 0, 7;
  8: 0, 0, 0, 0, 0, 0, 0, 8, 9;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 9;

%F Formula

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

%t Mathematica

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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from An interpolation operator, companion to A133080.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A133080 → D

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.