Real-rooted atlas

A103633 → 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
\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 17\,x^{2} + 14\,x^{3} + 4\,x^{4} - n - 7\,n x - 15\,n x^{2} - 13\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20123
300123
4001449
50001449
60001612827
700001612827

Show all 15 stored rows

Generated b-files: A056449 (row sums), A108411 (row sums), A162436 (row sums), A188440 ?

Computed in your browser ?

Technical details
Node id
n-b4a70761c59dc931 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-98cc1557e904900a

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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 0, 1;
  2: 0, 1, 2;
  3: 0, 0, 1, 2;
  4: 0, 0, 1, 4, 4;
  5: 0, 0, 0, 1, 4,  4;
  6: 0, 0, 0, 1, 6, 12,  8;
  7: 0, 0, 0, 0, 1,  6, 12,  8;
  8: 0, 0, 0, 0, 1,  8, 24, 32, 16;
  9: 0, 0, 0, 0, 0,  1,  8, 24, 32, 16;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-1+(7*n-8)*x+(15*n-17)*x^2+(13*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(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.

A188440: triangle, offset 0, from the row P0, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1, 2;
  3: 1, 2;
  4: 1, 4,  4;
  5: 1, 4,  4;
  6: 1, 6, 12,  8;
  7: 1, 6, 12,  8;
  8: 1, 8, 24, 32, 16;
  9: 1, 8, 24, 32, 16;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (1+2*x+x^2)*P(n,x) = -(n-1+(7*n-8)*x+(15*n-17)*x^2+(13*n-14)*x^3+(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 1 + (7*n - 8)*x + (15*n - 17)*x^2 + (13*n - 14)*x^3 + (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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-3c0198f97249d373
  • 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-3c0198f97249d373
  • 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-3c0198f97249d373

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 (10)

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