Real-rooted atlas

Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

A049218 (tail)

Proved for all n: all roots real · interlacing what this means

Recurrence from OEIS A049218
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(-n - n^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = -2 + x^{2}\)
\(P_{2}(x) = -8\,x + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1-201-1
20-801-7
3240-20015
401840-4001145
5-72007840-7001-5
60-8448024640-11201-6095
7403200-52352063840-16801-5815
806485760-2297600144480-24001433025

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-52650ae6398d7ffe · revision 1 · JSON
Definition source
oeis_catalog:A049218
Fingerprint
f6-3f77a25dcc8c4300

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 ?

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:       -2,         0,       1;
   2:        0,        -8,       0,        1;
   3:       24,         0,     -20,        0,       1;
   4:        0,       184,       0,      -40,       0,        1;
   5:     -720,         0,     784,        0,     -70,        0,     1;
   6:        0,     -8448,       0,     2464,       0,     -112,     0,     1;
   7:    40320,         0,  -52352,        0,    6384,        0,  -168,     0,    1;
   8:        0,    648576,       0,  -229760,       0,    14448,     0,  -240,    0,    1;
   9: -3628800,         0, 5360256,        0, -804320,        0, 29568,     0, -330,    0, 1;
  10:        0, -74972160,       0, 30633856,       0, -2393600,     0, 55968,    0, -440, 0, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) - (n^2+n)*P(n-2,x) for n >= 3, with P(1,x) = -2 + x^2, P(2,x) = -8*x + x^3.
%t Mathematica
P[1, x_] := -2 + x^2;
P[2, x_] := -8*x + x^3;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (n^2 + n)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = -2 + x^2; v[2] = -8*x + x^3; for(n = 3, N, my(i = n); v[i] = x*v[i - 1] - (n^2 + n)*v[i - 2]); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-ef0740a389023b53
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-ef0740a389023b53

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