Real-rooted atlas

A318772 → x ↦ (1/243)x → basis A013611

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-4}(x) + \left(3\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 9\)
\(P_{3}(x) = 27\)

Rows ?

nx0x1x2x3Pn(1)
011
133
299
32727
482486
524924273
6756108864
722954322727
869671628168611
921150590414427198
10642062084486485914
11194913721444320271377
125917062459281948864857186
131796268828288822247682707548
145453010276253233091257608552214

Computed in your browser ?

Technical details
Node id
n-720f52095a098452 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-56e5c46f7b07dbe7

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): no

OEIS matches ?

No OEIS matches.

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:     3;
  2:     9;
  3:    27;
  4:    82,    4;
  5:   249,   24;
  6:   756,  108;
  7:  2295,  432;
  8:  6967, 1628,  16;
  9: 21150, 5904, 144;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x)*P(n-4,x) + 3*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9, P(3,x) = 27.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[2, x_] := 9;
P[3, x_] := 27;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 4, x] + 3*P[n - 1, 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] = 3; v[3] = 9; v[4] = 27; for(n = 4, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 4] + 3*v[i - 1]); 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-8abfc1e62fa10a1f
  • 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-8abfc1e62fa10a1f
  • 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-8abfc1e62fa10a1f

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

No candidate equalities or related sequences.