Real-rooted atlas

A104684 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(1 - n\right)\,P_{n}(x) = \left(-2 - 4\,x - 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-2}(x) + \left(9 + 3\,x - 6\,n - 2\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3138122
46357151136
5321384156241886
61683250513803403515944
789891600811145380064548140636
848639100849849033706588551113631281488

Show all 15 stored rows

Generated b-files: A069835 (row sums) ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(-1 + n\right)\,P_{n}(x) = \left(-11 - 5\,x + 7\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-118f91023ad73cd8 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-0a47be968c31d18b)
Fingerprint
f6-c880512e8efe320e

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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,       1;
   3:      13,       8,       1;
   4:      63,      57,      15,       1;
   5:     321,     384,     156,      24,       1;
   6:    1683,    2505,    1380,     340,      35,      1;
   7:    8989,   16008,   11145,    3800,     645,     48,     1;
   8:   48639,  100849,   84903,   37065,    8855,   1113,    63,    1;
   9:  265729,  628736,  621040,  330848,  102340,  18368,  1792,   80,  1;
  10: 1462563, 3888657, 4408848, 2774352, 1055628, 247716, 34944, 2736, 99, 1;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) - (6*n-9+(2*n-3)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x] - (6*n - 9 + (2*n - 3)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2] - (6*n - 9 + (2*n - 3)*x)*v[i - 1])/(-n + 1)); 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-7b1b09c6fd205866
  • 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-7b1b09c6fd205866, e-da18ebef7a232130
  • 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-7b1b09c6fd205866, e-da18ebef7a232130
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-7b1b09c6fd205866, e-da18ebef7a232130
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-da18ebef7a232130

Proof chains

3 incoming proof steps. 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