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A346838 a(n) = (PolyLog(-n, -i) - exp(i*Pi*n)*PolyLog(-n, i)) * i / exp(i*Pi*n/2). 3
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 (list; graph; refs; listen; history; text; internal format)
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
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).
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
KEYWORD
sign
AUTHOR
Peter Luschny, Aug 12 2021
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)