Real-rooted atlas

A099097 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence
\(\left(-3 - 6\,x - 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(4 + 29\,x + 72\,x^{2} + 73\,x^{3} + 26\,x^{4} - 2\,n - 10\,n x - 27\,n x^{2} - 32\,n x^{3} - 13\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 25\,x^{2} + 51\,x^{3} + 43\,x^{4} + 13\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2033
3011011
40063339
500129109139
60009126360495
700015751611891763
8000012306203439276279

Show all 15 stored rows

Generated b-files: A007482 (row sums), A112906 ?

Computed in your browser ?

Technical details
Node id
n-6b91a68039c9d239 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-26728749caf1d6e4

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 -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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 3;
   3: 0, 1, 10;
   4: 0, 0,  6, 33;
   5: 0, 0,  1, 29, 109;
   6: 0, 0,  0,  9, 126, 360;
   7: 0, 0,  0,  1,  57, 516, 1189;
   8: 0, 0,  0,  0,  12, 306, 2034, 3927;
   9: 0, 0,  0,  0,   1,  94, 1491, 7807, 12970;
  10: 0, 0,  0,  0,   0,  15,  600, 6813, 29382, 42837;
%F Formula
The row polynomials satisfy (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n,x) = -(2*n-4+(10*n-29)*x+(27*n-72)*x^2+(32*n-73)*x^3+(13*n-26)*x^4)*P(n-1,x) + (4*x+25*x^2+51*x^3+43*x^4+13*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (10*n - 29)*x + (27*n - 72)*x^2 + (32*n - 73)*x^3 + (13*n - 26)*x^4)*P[n - 1, x] + (4*x + 25*x^2 + 51*x^3 + 43*x^4 + 13*x^5)*D[P[n - 1, x], x])/(3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 4 + (10*n - 29)*x + (27*n - 72)*x^2 + (32*n - 73)*x^3 + (13*n - 26)*x^4)*v[i - 1] + (4*x + 25*x^2 + 51*x^3 + 43*x^4 + 13*x^5)*deriv(v[i - 1], x))/(3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A112906: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 3;
  2: 0, 1, 10;
  3: 0, 0,  6, 33;
  4: 0, 0,  1, 29, 109;
  5: 0, 0,  0,  9, 126, 360;
  6: 0, 0,  0,  1,  57, 516, 1189;
  7: 0, 0,  0,  0,  12, 306, 2034, 3927;
  8: 0, 0,  0,  0,   1,  94, 1491, 7807, 12970;
  9: 0, 0,  0,  0,   0,  15,  600, 6813, 29382, 42837;
%F Formula
The row polynomials satisfy (3*n+6*n*x+3*n*x^2)*P(n,x) = -(2*n-2+(10*n-19)*x+(27*n-45)*x^2+(32*n-41)*x^3+(13*n-13)*x^4)*P(n-1,x) + (4*x+25*x^2+51*x^3+43*x^4+13*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[(-(2*n - 2 + (10*n - 19)*x + (27*n - 45)*x^2 + (32*n - 41)*x^3 + (13*n - 13)*x^4)*P[n - 1, x] + (4*x + 25*x^2 + 51*x^3 + 43*x^4 + 13*x^5)*D[P[n - 1, x], x])/(3*n + 6*n*x + 3*n*x^2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (-(2*n - 2 + (10*n - 19)*x + (27*n - 45)*x^2 + (32*n - 41)*x^3 + (13*n - 13)*x^4)*v[i - 1] + (4*x + 25*x^2 + 51*x^3 + 43*x^4 + 13*x^5)*deriv(v[i - 1], x))/(3*n + 6*n*x + 3*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, 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-a2968b8625ed90e3
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-a2968b8625ed90e3
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-a2968b8625ed90e3

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.

References

  1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741