Real-rooted atlas

A207823 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, −2/3) · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(16 + 69\,x + 111\,x^{2} + 79\,x^{3} + 21\,x^{4} - 8\,n - 43\,n x - 83\,n x^{2} - 69\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(12 + 68\,x + 153\,x^{2} + 171\,x^{3} + 95\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1459
215382477
356214272115657
42091068204317345515605
57804991127561627810372264047817
6291122378715911220041168165958412649407933
710864975163747367991901021600782754332864606053480129

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-5038739ef66fadea · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b0bdfa529c7b31c5

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 ?

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:      4,       5;
  2:     15,      38,      24;
  3:     56,     214,     272,      115;
  4:    209,    1068,    2043,     1734,      551;
  5:    780,    4991,   12756,    16278,    10372,     2640;
  6:   2911,   22378,   71591,   122004,   116816,    59584,    12649;
  7:  10864,   97516,  374736,   799190,  1021600,   782754,   332864,    60605;
  8:  40545,  416184, 1867266,  4782876,  7649935,  7823796,  4996593,  1821858,  290376;
  9: 151316, 1748217, 8969352, 26821656, 51519404, 65919715, 56185380, 30761310, 9816720, 1391275;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(8*n-16+(43*n-69)*x+(83*n-111)*x^2+(69*n-79)*x^3+(21*n-21)*x^4)*P(n-1,x) + (12+68*x+153*x^2+171*x^3+95*x^4+21*x^5)*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 - 16 + (43*n - 69)*x + (83*n - 111)*x^2 + (69*n - 79)*x^3 + (21*n - 21)*x^4)*P[n - 1, x] + (12 + 68*x + 153*x^2 + 171*x^3 + 95*x^4 + 21*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = (-(8*n - 16 + (43*n - 69)*x + (83*n - 111)*x^2 + (69*n - 79)*x^3 + (21*n - 21)*x^4)*v[i - 1] + (12 + 68*x + 153*x^2 + 171*x^3 + 95*x^4 + 21*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 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 - 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 [-1, -2/3)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-4baab92f0107de2d
  • 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-4baab92f0107de2d
  • 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-4baab92f0107de2d

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023