Real-rooted atlas

A099093 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(2\,n + 3\,n^{2} - 3\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(4 + 24\,x - 18\,x^{2} + 2\,n + 33\,n x - 18\,n x^{2} - 4\,n^{2} + 15\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 6\,x^{2} - 9\,x^{3} + 12\,n x - 3\,n x^{2} - 9\,n x^{3} + 4\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
101212
20128193
300162432594
40081129620253402
500012968100874818144
600043212150437403572192043
70000810087480214326139968449874

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e71ce56117a20184 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b1d4085ec10b350e

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 ?

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: 0, 12;
  2: 0, 12,  81;
  3: 0,  0, 162,  432;
  4: 0,  0,  81, 1296,  2025;
  5: 0,  0,   0, 1296,  8100,  8748;
  6: 0,  0,   0,  432, 12150, 43740,  35721;
  7: 0,  0,   0,    0,  8100, 87480, 214326,  139968;
  8: 0,  0,   0,    0,  2025, 87480, 535815,  979776,  531441;
  9: 0,  0,   0,    0,     0, 43740, 714420, 2939328, 4251528, 1968300;
%F Formula
The row polynomials satisfy (n^3+3*n^2+2*n-3*n^2*x)*P(n,x) = -(2*n^3+4*n^2-2*n-4-(15*n^2+33*n+24)*x+(18*n+18)*x^2)*P(n-1,x) + ((4*n^2+12*n+8)*x+(3*n^2-3*n-6)*x^2-(9*n+9)*x^3)*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 + 4*n^2 - 2*n - 4 - (15*n^2 + 33*n + 24)*x + (18*n + 18)*x^2)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (3*n^2 - 3*n - 6)*x^2 - (9*n + 9)*x^3)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n - 3*n^2*x)]];
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 + 4*n^2 - 2*n - 4 - (15*n^2 + 33*n + 24)*x + (18*n + 18)*x^2)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (3*n^2 - 3*n - 6)*x^2 - (9*n + 9)*x^3)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n - 3*n^2*x)); 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-4a001f0a65c0dd59
  • 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-4a001f0a65c0dd59
  • 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-4a001f0a65c0dd59

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.