Real-rooted atlas

A113953 → 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
\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 21\,x + 39\,x^{2} + 31\,x^{3} + 9\,x^{4} - 4\,n - 20\,n x - 37\,n x^{2} - 30\,n x^{3} - 9\,n x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 33\,x^{2} + 51\,x^{3} + 35\,x^{4} + 9\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20235
300459
4004141129
500012322165
60008488243181
700003213618885441
80000161443964381711165
9000008048010449843412929
1000000324001560271222026837589
110000001921520460867684852136519305
120000006410565440131521659610622273149661
1300000004484480176963590439836230565461126881
1400000001282688174725488095400942724976210923325525

Generated b-files: A006131 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-036ea97b0c6f24dd · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c33986a1af44d948

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): 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 2, 3;
  3: 0, 0, 4,  5;
  4: 0, 0, 4, 14, 11;
  5: 0, 0, 0, 12, 32,  21;
  6: 0, 0, 0,  8, 48,  82,  43;
  7: 0, 0, 0,  0, 32, 136, 188,   85;
  8: 0, 0, 0,  0, 16, 144, 396,  438, 171;
  9: 0, 0, 0,  0,  0,  80, 480, 1044, 984, 341;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(4*n-4+(20*n-21)*x+(37*n-39)*x^2+(30*n-31)*x^3+(9*n-9)*x^4)*P(n-1,x) + (8*x+33*x^2+51*x^3+35*x^4+9*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[(-(4*n - 4 + (20*n - 21)*x + (37*n - 39)*x^2 + (30*n - 31)*x^3 + (9*n - 9)*x^4)*P[n - 1, x] + (8*x + 33*x^2 + 51*x^3 + 35*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = (-(4*n - 4 + (20*n - 21)*x + (37*n - 39)*x^2 + (30*n - 31)*x^3 + (9*n - 9)*x^4)*v[i - 1] + (8*x + 33*x^2 + 51*x^3 + 35*x^4 + 9*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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-6bf6f9e9a94a64db
  • 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-6bf6f9e9a94a64db
  • 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-6bf6f9e9a94a64db

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 (1)

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

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [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