Real-rooted atlas

A267482 → x ↦ x+1

Proved for all n: all roots real, in (−3, 1) · interlacing (weak) what this means

Recurrence
\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(1 + n + n x\right)\,P_{n-1}(x) + \left(-3 + 2\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21315
3-13417
4-2-16519
5-1-61106111
61-6-116157113
721-18-1515218115
819-6-39-1529289117
9-1921-30-69-7493610119
10-2-13630-84-10514764511121
11-1-12149615-182-140541115512123
121-12-3480195-62-336-1621201556613125
1321-60-50260315-258-552-1532202097814127
14115-25-19015637393-648-825-883632749115129

Generated b-files: A005408 (row sums), A176271 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-35e99788be157971 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a130efc6d314ecdd

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): no
  • 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:  1,  3,   1;
  3: -1,  3,   4,   1;
  4: -2, -1,   6,   5,   1;
  5: -1, -6,   1,  10,   6,  1;
  6:  1, -6, -11,   6,  15,  7,  1;
  7:  2,  1, -18, -15,  15, 21,  8,  1;
  8:  1,  9,  -6, -39, -15, 29, 28,  9,  1;
  9: -1,  9,  21, -30, -69, -7, 49, 36, 10, 1;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (n+1+n*x)*P(n-1,x) - (3-2*x-x^2)*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[((n + 1 + n*x)*P[n - 1, x] - (3 - 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((n + 1 + n*x)*v[i - 1] - (3 - 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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 (-3, 1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-4fb0c58de9dbd95f
  • 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-4fb0c58de9dbd95f

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