A152830 → basis A013609

palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

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

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

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

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

Rows ?

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

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-438a1c18c8e36249 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-a2790537ca9ffc7c

Observed on the stored rows ?

Degrees: 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3 (irregular)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A242954: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

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

%F Formula

The row polynomials satisfy P(n,x) = (2+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[(2 + 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] = (2 + 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8e5544e9659d3042
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-8e5544e9659d3042
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-8e5544e9659d3042

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013609 e-8e5544e9659d3042

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, A007318 rows repeated four times .
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-0c1eb1fd87e22444 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A152830: Triangle read by rows, A007318 rows repeated four times .