Real-rooted atlas

A063967 → 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
\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(15 + 68\,x + 115\,x^{2} + 86\,x^{3} + 24\,x^{4} - 7\,n - 32\,n x - 55\,n x^{2} - 42\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 31\,x + 75\,x^{2} + 89\,x^{3} + 52\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
327615
4316281663
55359110444279
68702434183561201215
7131365891351173011723285319
8212561330381665286656374489623247

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f263ca9517b1fbce · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-261a981b55e1ac6d

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 ?

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:  1,   2;
   3:  2,   7,    6;
   4:  3,  16,   28,    16;
   5:  5,  35,   91,   104,    44;
   6:  8,  70,  243,   418,   356,    120;
   7: 13, 136,  589,  1351,  1730,   1172,    328;
   8: 21, 256, 1330,  3816,  6528,   6656,   3744,   896;
   9: 34, 473, 2865,  9866, 21121,  28776,  24360, 11712,  2448;
  10: 55, 860, 5951, 23914, 61489, 104890, 118680, 85872, 36048, 6688;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(7*n-15+(32*n-68)*x+(55*n-115)*x^2+(42*n-86)*x^3+(12*n-24)*x^4)*P(n-1,x) + (5+31*x+75*x^2+89*x^3+52*x^4+12*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[(-(7*n - 15 + (32*n - 68)*x + (55*n - 115)*x^2 + (42*n - 86)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*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] = (-(7*n - 15 + (32*n - 68)*x + (55*n - 115)*x^2 + (42*n - 86)*x^3 + (12*n - 24)*x^4)*v[i - 1] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*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.

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-5274886585caff6b
  • 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-5274886585caff6b
  • 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-5274886585caff6b

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. [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