Real-rooted atlas

A152828 → D+1

nonnegative coefficients only rational roots

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

Recurrence
\(\left(54 - 21\,x - 108\,x^{2} + 27\,x^{3} - 25\,n - 45\,n x + 27\,n x^{2} - 3\,n^{2} + 9\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(168 + 12\,x - 237\,x^{2} - 54\,x^{3} + 27\,x^{4} - 97\,n - 115\,n x + 9\,n x^{2} + 27\,n x^{3} + 6\,n^{2} + 15\,n^{2} x + 9\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-3}(x) + \left(162 + 297\,x + 108\,x^{2} - 27\,x^{3} - 27\,n - 54\,n x - 27\,n x^{2}\right)\,P_{n-3}'(x) + \left(27 + 81\,x + 81\,x^{2} + 27\,x^{3}\right)\,P_{n-3}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2Pn(1)
011
111
211
3213
4213
5213
63418
73418

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-39d43b83afedcf47 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a4bd419f5eb6b3d0

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 (irregular)
  • 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): yes

OEIS matches ?

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 2, 1;
  4: 2, 1;
  5: 2, 1;
  6: 3, 4, 1;
  7: 3, 4, 1;
  8: 3, 4, 1;
  9: 4, 9, 6, 1;
%F Formula
The row polynomials satisfy (n^3-3*n^2-25*n+54+(9*n^2-45*n-21)*x+(27*n-108)*x^2+27*x^3)*P(n,x) = (n^3+6*n^2-97*n+168+(n^3+15*n^2-115*n+12)*x+(9*n^2+9*n-237)*x^2+(27*n-54)*x^3+27*x^4)*P(n-3,x) - (27*n-162+(54*n-297)*x+(27*n-108)*x^2+27*x^3)*d/dx P(n-3,x) + (27+81*x+81*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 6*n^2 - 97*n + 168 + (n^3 + 15*n^2 - 115*n + 12)*x + (9*n^2 + 9*n - 237)*x^2 + (27*n - 54)*x^3 + 27*x^4)*P[n - 3, x] - (27*n - 162 + (54*n - 297)*x + (27*n - 108)*x^2 + 27*x^3)*D[P[n - 3, x], x] + (27 + 81*x + 81*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 - 3*n^2 - 25*n + 54 + (9*n^2 - 45*n - 21)*x + (27*n - 108)*x^2 + 27*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; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 6*n^2 - 97*n + 168 + (n^3 + 15*n^2 - 115*n + 12)*x + (9*n^2 + 9*n - 237)*x^2 + (27*n - 54)*x^3 + 27*x^4)*v[i - 3] - (27*n - 162 + (54*n - 297)*x + (27*n - 108)*x^2 + 27*x^3)*deriv(v[i - 3], x) + (27 + 81*x + 81*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 - 3*n^2 - 25*n + 54 + (9*n^2 - 45*n - 21)*x + (27*n - 108)*x^2 + 27*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-809cfb1679ce14dc
  • 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-809cfb1679ce14dc
  • 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-809cfb1679ce14dc

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