Real-rooted atlas

A063967 → θ+1

nonnegative coefficients

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

Recurrence
\(\left(-10 - 8\,x + 10\,n + 3\,n x - 2\,n x^{2} + 5\,n^{2} x + n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(50 + 40\,x + 6\,x^{2} - 20\,n + 29\,n x + 25\,n x^{2} + 4\,n x^{3} - 15\,n^{2} x + n^{2} x^{2} + 2\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 45\,x + 23\,x^{2} + 3\,x^{3} + 25\,n x + 40\,n x^{2} + 17\,n x^{3} + 2\,n x^{4} + 5\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
326311
431415436
553048285116
6860129116456364
7131163123802306671124
82121870510848954029183424

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-363260c438af1335 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8b51e30b3c5c22f5

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 ?

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,   2;
   3:  2,   6,    3;
   4:  3,  14,   15,    4;
   5:  5,  30,   48,   28,    5;
   6:  8,  60,  129,  116,   45,    6;
   7: 13, 116,  312,  380,  230,   66,    7;
   8: 21, 218,  705, 1084,  895,  402,   91,   8;
   9: 34, 402, 1518, 2820, 2955, 1818,  644, 120,   9;
  10: 55, 730, 3153, 6868, 8730, 6840, 3325, 968, 153, 10;
%F Formula
The row polynomials satisfy (10*n-10+(5*n^2+3*n-8)*x+(n^3+n^2-2*n)*x^2)*P(n,x) = -(20*n-50+(15*n^2-29*n-40)*x+(2*n^3-n^2-25*n-6)*x^2-(2*n^2+4*n)*x^3)*P(n-1,x) + (25+(25*n+45)*x+(5*n^2+40*n+23)*x^2+(6*n^2+17*n+3)*x^3+(n^2+2*n)*x^4)*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[(-(20*n - 50 + (15*n^2 - 29*n - 40)*x + (2*n^3 - n^2 - 25*n - 6)*x^2 - (2*n^2 + 4*n)*x^3)*P[n - 1, x] + (25 + (25*n + 45)*x + (5*n^2 + 40*n + 23)*x^2 + (6*n^2 + 17*n + 3)*x^3 + (n^2 + 2*n)*x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 + 3*n - 8)*x + (n^3 + n^2 - 2*n)*x^2)]];
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] = (-(20*n - 50 + (15*n^2 - 29*n - 40)*x + (2*n^3 - n^2 - 25*n - 6)*x^2 - (2*n^2 + 4*n)*x^3)*v[i - 1] + (25 + (25*n + 45)*x + (5*n^2 + 40*n + 23)*x^2 + (6*n^2 + 17*n + 3)*x^3 + (n^2 + 2*n)*x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 + 3*n - 8)*x + (n^3 + n^2 - 2*n)*x^2)); 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-bf6c3a69738a7962
  • 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-bf6c3a69738a7962
  • 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-bf6c3a69738a7962

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