Real-rooted atlas

A135838 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 26\,x^{2} + 24\,x^{3} + 8\,x^{4}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
228818
34244832108
44329612864324
58803206406402561944
68964801280192015365125832
716224134444808960107527168204834992

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e8ae0bac7c0efceb · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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

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:  2,   4;
  2:  2,   8,    8;
  3:  4,  24,   48,    32;
  4:  4,  32,   96,   128,    64;
  5:  8,  80,  320,   640,   640,    256;
  6:  8,  96,  480,  1280,  1920,   1536,    512;
  7: 16, 224, 1344,  4480,  8960,  10752,   7168,   2048;
  8: 16, 256, 1792,  7168, 17920,  28672,  28672,  16384,  4096;
  9: 32, 576, 4608, 21504, 64512, 129024, 172032, 147456, 73728, 16384;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+12*x+26*x^2+24*x^3+8*x^4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 12*x + 26*x^2 + 24*x^3 + 8*x^4)*P[n - 2, x])/(1 + 2*x + x^2)]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2 + 12*x + 26*x^2 + 24*x^3 + 8*x^4)*v[i - 2])/(1 + 2*x + x^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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-9f3b1e4aefe8044e
  • 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-9f3b1e4aefe8044e
  • 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-9f3b1e4aefe8044e

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

Same fingerprint (67)

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