Real-rooted atlas

A274498 → 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
\(\left(-2 - 6\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(2 - 30\,x + 24\,x^{2} - 8\,x^{3} - 19\,n + 34\,n x - 20\,n x^{2} + 8\,n x^{3} + 13\,n^{2} - 12\,n^{2} x + 8\,n^{2} x^{2} - 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 12\,x^{2} - 16\,x^{4} - 4\,n - 2\,n x + 4\,n x^{2} - 16\,n x^{3} + 2\,n^{2} + 2\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
29615
3271845
44510836189
5135324108567
61898109722162187
7567243029166486561
8729453697207776129624057
92187136082916023328388872171
102673218706804097200583207776255879
1180196561020412029160017496023328767637
12947796228393660816480874800419904466562657205
132843128868411809802449440262440012597121399687971615
14328053980342020788551124085730407348320293932827993627103491

Computed in your browser ?

Technical details
Node id
n-6dbf222e2d76f379 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4f4ca2dec4972ed6

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): 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:    3;
  2:    9,     6;
  3:   27,    18;
  4:   45,   108,    36;
  5:  135,   324,   108;
  6:  189,   810,   972,   216;
  7:  567,  2430,  2916,   648;
  8:  729,  4536,  9720,  7776, 1296;
  9: 2187, 13608, 29160, 23328, 3888;
%F Formula
The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = -(2*n^3-13*n^2+19*n-2-(2*n^3-12*n^2+34*n-30)*x-(8*n^2-20*n+24)*x^2-(8*n-8)*x^3)*P(n-1,x) + (2*n^2-4*n+(2*n^2-2*n-8)*x-(4*n^2-4*n+12)*x^2-16*n*x^3-16*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 - 13*n^2 + 19*n - 2 - (2*n^3 - 12*n^2 + 34*n - 30)*x - (8*n^2 - 20*n + 24)*x^2 - (8*n - 8)*x^3)*P[n - 1, x] + (2*n^2 - 4*n + (2*n^2 - 2*n - 8)*x - (4*n^2 - 4*n + 12)*x^2 - 16*n*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*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 - 13*n^2 + 19*n - 2 - (2*n^3 - 12*n^2 + 34*n - 30)*x - (8*n^2 - 20*n + 24)*x^2 - (8*n - 8)*x^3)*v[i - 1] + (2*n^2 - 4*n + (2*n^2 - 2*n - 8)*x - (4*n^2 - 4*n + 12)*x^2 - 16*n*x^3 - 16*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (4*n - 6)*x + 4*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-dfefcaa0f678161b
  • 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-dfefcaa0f678161b
  • 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-dfefcaa0f678161b

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

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