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

Author

Peter Luschny, Aug 12 2021

Keywords

• sign

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.

Crossrefs

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

Programs

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

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}]

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