Real-rooted atlas

A128099 → D+1

nonnegative coefficients

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

Recurrence
\(\left(3\,n - 19\,n x + 8\,n x^{2} - 5\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(44\,x^{2} - 32\,x^{3} + n + n x - 68\,n x^{2} + 32\,n x^{3} - 3\,n^{2} - 20\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(3\,x^{2} + 16\,x^{3} - 64\,x^{4} + n x - 64\,n x^{3} - 2\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2325
3549
4714425
59321253
61158488125
7139213632273
81513430014416609
917184564480801325
10192429521240400322885
11213081488268815201926217
122338221965152448010566413353
13254643100902410976448044828517
14275544224147602352014784268812860685

Generated b-files: A127976 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-163abed26b768e53 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f319c3e41f65e203

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:  1;
  2:  3,   2;
  3:  5,   4;
  4:  7,  14,   4;
  5:  9,  32,  12;
  6: 11,  58,  48,   8;
  7: 13,  92, 136,  32;
  8: 15, 134, 300, 144, 16;
  9: 17, 184, 564, 480, 80;
%F Formula
The row polynomials satisfy (2*n^3-5*n^2+3*n+(8*n^2-19*n)*x+8*n*x^2)*P(n,x) = (2*n^3-3*n^2+n+(8*n^3-20*n^2+n)*x+(32*n^2-68*n+44)*x^2+(32*n-32)*x^3)*P(n-1,x) - ((2*n^2-n)*x+(16*n^2-3)*x^2+(64*n-16)*x^3+64*x^4)*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 - 3*n^2 + n + (8*n^3 - 20*n^2 + n)*x + (32*n^2 - 68*n + 44)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] - ((2*n^2 - n)*x + (16*n^2 - 3)*x^2 + (64*n - 16)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 5*n^2 + 3*n + (8*n^2 - 19*n)*x + 8*n*x^2)]];
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 - 3*n^2 + n + (8*n^3 - 20*n^2 + n)*x + (32*n^2 - 68*n + 44)*x^2 + (32*n - 32)*x^3)*v[i - 1] - ((2*n^2 - n)*x + (16*n^2 - 3)*x^2 + (64*n - 16)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 5*n^2 + 3*n + (8*n^2 - 19*n)*x + 8*n*x^2)); 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-0ace6e908f8ea6d4
  • 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-0ace6e908f8ea6d4
  • 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-0ace6e908f8ea6d4

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.