Real-rooted atlas

A054142 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-4 + 13\,x - 4\,x^{2} + 16\,n - 34\,n x + 8\,n x^{2} - 20\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(10\,x + 41\,x^{2} - 20\,x^{3} + 4\,n - 8\,n x - 86\,n x^{2} + 24\,n x^{3} - 12\,n^{2} - 36\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(5\,x^{2} + 16\,x^{3} - 16\,x^{4} + 2\,n x - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
245110
36179133
483745141105
5106513395201324
612101297364175271977
71414556110058402943512895

Show all 15 stored rows

Generated b-files: A027989 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-9887e767efe7cad8 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-285a8f351a67f82b

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 ?

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:  2,   1;
  2:  4,   5,    1;
  3:  6,  17,    9,    1;
  4:  8,  37,   45,   14,    1;
  5: 10,  65,  133,   95,   20,    1;
  6: 12, 101,  297,  364,  175,   27,    1;
  7: 14, 145,  561, 1005,  840,  294,   35,   1;
  8: 16, 197,  949, 2266, 2805, 1722,  462,  44,  1;
  9: 18, 257, 1485, 4459, 7436, 6831, 3234, 690, 54, 1;
%F Formula
The row polynomials satisfy (8*n^3-20*n^2+16*n-4+(16*n^2-34*n+13)*x+(8*n-4)*x^2)*P(n,x) = (8*n^3-12*n^2+4*n+(24*n^3-36*n^2-8*n+10)*x+(48*n^2-86*n+41)*x^2+(24*n-20)*x^3)*P(n-1,x) - ((4*n^2-2*n)*x+(16*n^2-5)*x^2+(32*n-16)*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[((8*n^3 - 12*n^2 + 4*n + (24*n^3 - 36*n^2 - 8*n + 10)*x + (48*n^2 - 86*n + 41)*x^2 + (24*n - 20)*x^3)*P[n - 1, x] - ((4*n^2 - 2*n)*x + (16*n^2 - 5)*x^2 + (32*n - 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 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] = ((8*n^3 - 12*n^2 + 4*n + (24*n^3 - 36*n^2 - 8*n + 10)*x + (48*n^2 - 86*n + 41)*x^2 + (24*n - 20)*x^3)*v[i - 1] - ((4*n^2 - 2*n)*x + (16*n^2 - 5)*x^2 + (32*n - 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^2 - 34*n + 13)*x + (8*n - 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-f4e836bf3f81ac68
  • 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-f4e836bf3f81ac68
  • 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-f4e836bf3f81ac68
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f4e836bf3f81ac68

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023