Real-rooted atlas

A026729 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-2 + 8\,x - x^{3} + 5\,n - 11\,n x - n x^{2} + n x^{3} - 4\,n^{2} + 3\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(16 - 28\,x - 4\,x^{2} + 5\,x^{3} + x^{4} - 28\,n + 19\,n x + 6\,n x^{2} - n x^{3} + 14\,n^{2} - 3\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 18\,x^{2} + 3\,x^{3} + 6\,x^{4} + x^{5} - 12\,n x + 9\,n x^{2} + 7\,n x^{3} + n x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31315
4045110
502107120
6009199138
7003253111171
8000165446131130

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-4b52398bfaf800c9 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-01a1ce136819b8d3

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, 1;
   3: 1, 3,  1;
   4: 0, 4,  5,  1;
   5: 0, 2, 10,  7,  1;
   6: 0, 0,  9, 19,  9,   1;
   7: 0, 0,  3, 25, 31,  11,   1;
   8: 0, 0,  0, 16, 54,  46,  13,  1;
   9: 0, 0,  0,  4, 51, 100,  64, 15,  1;
  10: 0, 0,  0,  0, 25, 125, 167, 85, 17, 1;
%F Formula
The row polynomials satisfy (n^3-4*n^2+5*n-2+(3*n^2-11*n+8)*x+(n^2-n)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16+(3*n^2-19*n+28)*x+(2*n^2-6*n+4)*x^2+(n-5)*x^3-x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+9*n-18)*x^2+(7*n+3)*x^3+(n+6)*x^4+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^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 19*n + 28)*x + (2*n^2 - 6*n + 4)*x^2 + (n - 5)*x^3 - x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 18)*x^2 + (7*n + 3)*x^3 + (n + 6)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 5*n - 2 + (3*n^2 - 11*n + 8)*x + (n^2 - n)*x^2 + (n - 1)*x^3)]];
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^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 19*n + 28)*x + (2*n^2 - 6*n + 4)*x^2 + (n - 5)*x^3 - x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 18)*x^2 + (7*n + 3)*x^3 + (n + 6)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 5*n - 2 + (3*n^2 - 11*n + 8)*x + (n^2 - n)*x^2 + (n - 1)*x^3)); 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-ab9bf473034a482c
  • 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-ab9bf473034a482c
  • 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-ab9bf473034a482c

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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.