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%I #28 Sep 08 2022 08:44:37
%S 1,0,1,-3,8,-30,136,-693,3968,-25260,176896,-1351383,11184128,
%T -99680490,951878656,-9695756073,104932671488,-1202439837720,
%U 14544442556416,-185185594118763,2475749026562048,-34674437196568950
%N Expansion of e.g.f.: cosh(log(1+sin(x))).
%C |a(n)| = number of even alternating permutations on n letters (offset 1). - _Vladeta Jovovic_, May 20 2007
%H G. C. Greubel, <a href="/A009123/b009123.txt">Table of n, a(n) for n = 0..250</a>
%F a(2*n) = (1/2)*A000182(n+1); a(2*n+1) = A012007(n+1) = A009567(2*n+1) + 1.
%F G.f.: (1+x/(1+x^2))/2 + 1/2/Q(0) where Q(k) = 1 + (k+1)*x - x^2*(k+1)*(k+2)/2 /Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Mar 12 2013
%F a(n) ~ n! * n * (-1)^n * (2/Pi)^(n+2). - _Vaclav Kotesovec_, Jan 22 2015
%t CoefficientList[Series[(1 + (1 + Sin[x])^2)/(2*(1 + Sin[x])), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 22 2015 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace(cosh(log(1+sin(x))))) \\ _G. C. Greubel_, Jul 26 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(Log(1+Sin(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 26 2018
%Y Cf. A000111.
%K sign,easy
%O 0,4
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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