A346838 a(n) = (PolyLog(-n, -i) - exp(i*Pi*n)*PolyLog(-n, i)) * i / exp(i*Pi*n/2).

1, -1, 1, -2, 5, -16, 61, -272, 1385, -7936, 50521, -353792, 2702765, -22368256, 199360981, -1903757312, 19391512145, -209865342976, 2404879675441, -29088885112832, 370371188237525, -4951498053124096, 69348874393137901, -1015423886506852352, 15514534163557086905

Offset

0,4

Comments

This is a signed variant of A000111. The author named the interpolating function of A000111 the 'André function' and the interpolating function of this sequence the 'signed André function'. See the illustrating file in the links section for the definitions.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..485

Désiré André, Développement de sec x and tan x, C. R. Math. Acad. Sci. Paris, Vol. 88 (1879), pp. 965-979.

Désiré André, Mémoire sur les permutations alternées, J. Math. Pur. Appl., 7, 167-184, (1881).

Peter Luschny, Illustrating the André function.

R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240 [math.CO], 2009.

Formula

log(abs(a(n))) = log(A000111(n)) ~ log(4) + (1/2 + n)*log(2*n/Pi) + ((2/7) - n^2 + 30*n^4 - 360*n^6) / (360*n^5).

E.g.f.: sec(x) - tan(x). - Ilya Gutkovskiy, Aug 12 2021

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o+j-1, u-j), j=1..u))
    end:
a:= n-> (-1)^n*b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 05 2021

Mathematica

a[n_] := I (PolyLog[-n, -I] - Exp[I Pi n] PolyLog[-n, I]) / Exp[I Pi n / 2];
Table[a[n], {n, 0, 24}]

Prog

(Julia)
using Nemo
CC = ComplexField(80); I = onei(CC); Pi = const_pi(CC)
A(n) = I*(polylog(-n, -I) - exp(I*Pi*n)*polylog(-n, I)) / exp(I*Pi*n/CC(2))
[unique_integer(A(CC(n)))[2] for n in 0:24] |> println

Crossrefs

Cf. A000111 (unsigned version), A346839 (infinite sum).

Sequence in context: A138265 A275711 A163747 * A000111 A007976 A058259

Adjacent sequences: A346835 A346836 A346837 * A346839 A346840 A346841

Keyword

sign

Author

Peter Luschny, Aug 12 2021

Status

approved