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%I #24 Oct 05 2021 18:59:55
%S 1,-1,1,-2,5,-16,61,-272,1385,-7936,50521,-353792,2702765,-22368256,
%T 199360981,-1903757312,19391512145,-209865342976,2404879675441,
%U -29088885112832,370371188237525,-4951498053124096,69348874393137901,-1015423886506852352,15514534163557086905
%N a(n) = (PolyLog(-n, -i) - exp(i*Pi*n)*PolyLog(-n, i)) * i / exp(i*Pi*n/2).
%C 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.
%H Alois P. Heinz, <a href="/A346838/b346838.txt">Table of n, a(n) for n = 0..485</a>
%H Désiré André, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k30457/f961.item">Développement de sec x and tan x</a>, C. R. Math. Acad. Sci. Paris, Vol. 88 (1879), pp. 965-979.
%H Désiré André, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1881_3_7_A10_0.pdf">Mémoire sur les permutations alternées</a>, J. Math. Pur. Appl., 7, 167-184, (1881).
%H Peter Luschny, <a href="/A346838/a346838.pdf">Illustrating the André function.</a>
%H R. P. Stanley, <a href="https://arxiv.org/abs/0912.4240">A survey of alternating permutations</a>, arXiv:0912.4240 [math.CO], 2009.
%F 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).
%F E.g.f.: sec(x) - tan(x). - _Ilya Gutkovskiy_, Aug 12 2021
%p b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p add(b(o+j-1, u-j), j=1..u))
%p end:
%p a:= n-> (-1)^n*b(n, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 05 2021
%t a[n_] := I (PolyLog[-n, -I] - Exp[I Pi n] PolyLog[-n, I]) / Exp[I Pi n / 2];
%t Table[a[n], {n, 0, 24}]
%o (Julia)
%o using Nemo
%o CC = ComplexField(80); I = onei(CC); Pi = const_pi(CC)
%o A(n) = I*(polylog(-n, -I) - exp(I*Pi*n)*polylog(-n, I)) / exp(I*Pi*n/CC(2))
%o [unique_integer(A(CC(n)))[2] for n in 0:24] |> println
%Y Cf. A000111 (unsigned version), A346839 (infinite sum).
%K sign
%O 0,4
%A _Peter Luschny_, Aug 12 2021
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