This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346838 #24 Oct 05 2021 18:59:55
%S A346838 1,-1,1,-2,5,-16,61,-272,1385,-7936,50521,-353792,2702765,-22368256,
%T A346838 199360981,-1903757312,19391512145,-209865342976,2404879675441,
%U A346838 -29088885112832,370371188237525,-4951498053124096,69348874393137901,-1015423886506852352,15514534163557086905
%N A346838 a(n) = (PolyLog(-n, -i) - exp(i*Pi*n)*PolyLog(-n, i)) * i / exp(i*Pi*n/2).
%C A346838 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 A346838 Alois P. Heinz, <a href="/A346838/b346838.txt">Table of n, a(n) for n = 0..485</a>
%H A346838 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 A346838 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 A346838 Peter Luschny, <a href="/A346838/a346838.pdf">Illustrating the André function.</a>
%H A346838 R. P. Stanley, <a href="https://arxiv.org/abs/0912.4240">A survey of alternating permutations</a>, arXiv:0912.4240 [math.CO], 2009.
%F A346838 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 A346838 E.g.f.: sec(x) - tan(x). - _Ilya Gutkovskiy_, Aug 12 2021
%p A346838 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A346838 add(b(o+j-1, u-j), j=1..u))
%p A346838 end:
%p A346838 a:= n-> (-1)^n*b(n, 0):
%p A346838 seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 05 2021
%t A346838 a[n_] := I (PolyLog[-n, -I] - Exp[I Pi n] PolyLog[-n, I]) / Exp[I Pi n / 2];
%t A346838 Table[a[n], {n, 0, 24}]
%o A346838 (Julia)
%o A346838 using Nemo
%o A346838 CC = ComplexField(80); I = onei(CC); Pi = const_pi(CC)
%o A346838 A(n) = I*(polylog(-n, -I) - exp(I*Pi*n)*polylog(-n, I)) / exp(I*Pi*n/CC(2))
%o A346838 [unique_integer(A(CC(n)))[2] for n in 0:24] |> println
%Y A346838 Cf. A000111 (unsigned version), A346839 (infinite sum).
%K A346838 sign
%O A346838 0,4
%A A346838 _Peter Luschny_, Aug 12 2021