Real-rooted atlas

A099093 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-3\,n x - 5\,n x^{2} + 3\,n x^{3} - n^{2} + n^{2} x + 3\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 16\,x^{2} + 9\,x^{4} - 6\,n - n x + 10\,n x^{2} + 3\,n x^{3} + 8\,n^{2} + 7\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} - 2\,x^{3} + 15\,x^{4} + 9\,x^{5} - 4\,n x + n x^{2} + 15\,n x^{3} + 9\,n x^{4} + 4\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2321933
30369927162
401825240581756
500243137715392433402
6008119716561558972914931
70001296124742867419683218764314

Show all 15 stored rows

Generated b-files: A099432 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-5240258528480355 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-441b88ccebee6207

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: 3,  3;
  2: 3, 21,   9;
  3: 0, 36,  99,   27;
  4: 0, 18, 252,  405,    81;
  5: 0,  0, 243, 1377,  1539,   243;
  6: 0,  0,  81, 1971,  6561,  5589,    729;
  7: 0,  0,   0, 1296, 12474, 28674,  19683,   2187;
  8: 0,  0,   0,  324, 12231, 68040, 118098,  67797,   6561;
  9: 0,  0,   0,    0,  6075, 88695, 336069, 465831, 229635, 19683;
%F Formula
The row polynomials satisfy (n^3-n^2+(n^2-3*n)*x+(3*n^2-5*n)*x^2+3*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(7*n^2-n-12)*x+(6*n^2-10*n+16)*x^2-3*n*x^3-9*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(3*n^2+n-8)*x^2+(15*n-2)*x^3+(9*n+15)*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[(-(2*n^3 - 8*n^2 + 6*n - (7*n^2 - n - 12)*x + (6*n^2 - 10*n + 16)*x^2 - 3*n*x^3 - 9*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (3*n^2 + n - 8)*x^2 + (15*n - 2)*x^3 + (9*n + 15)*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (n^2 - 3*n)*x + (3*n^2 - 5*n)*x^2 + 3*n*x^3)]];
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] = (-(2*n^3 - 8*n^2 + 6*n - (7*n^2 - n - 12)*x + (6*n^2 - 10*n + 16)*x^2 - 3*n*x^3 - 9*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (3*n^2 + n - 8)*x^2 + (15*n - 2)*x^3 + (9*n + 15)*x^4 + 9*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 + (n^2 - 3*n)*x + (3*n^2 - 5*n)*x^2 + 3*n*x^3)); 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b19f0f91a9476669
  • 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-b19f0f91a9476669
  • 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-b19f0f91a9476669

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