Real-rooted atlas

A102547 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0112
11214
213318
32795124
431322187164
542145503091160
66369112510046111416
7961178291290179661311088

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-634d40a3ea7efaff · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8d0025ba69fbd5cb

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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

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;
  1:  1,   2,   1;
  2:  1,   3,   3,    1;
  3:  2,   7,   9,    5,    1;
  4:  3,  13,  22,   18,    7,    1;
  5:  4,  21,  45,   50,   30,    9,    1;
  6:  6,  36,  91,  125,  100,   46,   11,   1;
  7:  9,  61, 178,  291,  290,  179,   66,  13,   1;
  8: 13,  99, 330,  630,  756,  588,  294,  90,  15,  1;
  9: 19, 160, 598, 1303, 1827, 1715, 1085, 453, 118, 17, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-3,x) + (1+x)*P(n-1,x) for n >= 3, with P(0,x) = 1 + x, P(1,x) = 1 + 2*x + x^2, P(2,x) = 1 + 3*x + 3*x^2 + x^3.
%t Mathematica
P[0, x_] := 1 + x;
P[1, x_] := 1 + 2*x + x^2;
P[2, x_] := 1 + 3*x + 3*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 3, x] + (1 + x)*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 + x; v[2] = 1 + 2*x + x^2; v[3] = 1 + 3*x + 3*x^2 + x^3; for(n = 3, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 3] + (1 + x)*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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-1dde996805ab9605
  • 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-1dde996805ab9605, e-3d86f7db70abac1a
  • 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-1dde996805ab9605, e-3d86f7db70abac1a
  • 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-3d86f7db70abac1a

Proof chains

2 incoming proof steps. 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.