Real-rooted atlas

A126216 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 4\,x - 2\,x^{2} - 8\,x^{3} - 4\,x^{4} + 4\,n + 17\,n x + 26\,n x^{2} + 17\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
25151131
314639345215
4422525605461971597
513299029404320314090312425
6429386114355282153091517901427999955
71430150156706716516524216521136510167520793824675

Show all 15 stored rows

Generated b-files: A269730 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-ff3f26cc45b3d80a · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ca6d936c14642db2

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,      3;
  2:     5,     15,      11;
  3:    14,     63,      93,      45;
  4:    42,    252,     560,     546,      197;
  5:   132,    990,    2940,    4320,     3140,      903;
  6:   429,   3861,   14355,   28215,    30915,    17901,     4279;
  7:  1430,  15015,   67067,  165165,   242165,   211365,   101675,   20793;
  8:  4862,  58344,  304304,  900900,  1655654,  1933932,  1402016,  576708,  103049;
  9: 16796, 226746, 1352520, 4678128, 10339056, 15140034, 14688492, 9103536, 3270438, 518859;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (4*n+2+(17*n+4)*x+(26*n-2)*x^2+(17*n-8)*x^3+(4*n-4)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*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[((4*n + 2 + (17*n + 4)*x + (26*n - 2)*x^2 + (17*n - 8)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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] = ((4*n + 2 + (17*n + 4)*x + (26*n - 2)*x^2 + (17*n - 8)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-38a5f21f7e3f9e41
  • 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-38a5f21f7e3f9e41
  • 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-38a5f21f7e3f9e41

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. [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