%I M1523 #44 Aug 17 2018 13:41:40

%S 1,1,1,2,5,20,85,520,3145,26000,204425,2132000,20646925,260104000,

%T 2993804125,44217680000,589779412625,9993195680000,151573309044625,

%U 2898026747200000,49261325439503125,1049085682486400000,19753791501240753125,463695871658988800000

%N Expansion of exp(arcsin(x)).

%D L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006228/b006228.txt">Table of n, a(n) for n = 0..100</a>

%H H. S. Uhler, <a href="http://www.jstor.org/stable/2972387">On the numerical value of i^i</a>, Amer. Math. Monthly, 28 (1921), 114-116.

%H Kruchinin Vladimir Victorovich, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F i even: a_i = Product_{j=1..i/2-1} 1 + 4j^2, i odd: a_i = Product_{j=1..(i-1)/2} 2 + 4j(j-1). - Cris Moore (moore(AT)santafe.edu), Jan 31 2001

%F a(0)=1, a(1)=1, a(n) = (1+(n-2)^2)*a(n-2) for n >= 2. _Jaume Oliver Lafont_, Oct 24 2009

%F a(n) = (n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(C(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*C(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!, i=0..floor(j/2))*(-1)^(k-j), j=1..k))*C(k+n-1,n-1), k=1..n-m))/(m-1)!, m=1..n), n>0. - _Vladimir Kruchinin_, Sep 12 2010

%F E.g.f.: exp(arcsin(x))=1+2z/(H(0)-z); H(k)=4k+2+z^2*(4k^2+8k+5)/H(k+1), where z=x/((1-x^2)^1/2); (continued fraction). - _Sergei N. Gladkovskii_, Nov 20 2011

%F a(n) ~ (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 23 2013

%F a(n) = 2^(n-2) * (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * GAMMA((n-I)/2) * GAMMA((n+I)/2) / Pi. - _Vaclav Kotesovec_, Nov 06 2014

%p a:= n-> n!*coeff(series(exp(arcsin(x)), x, n+1), x, n):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 17 2018

%t Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (* _Robert G. Wilson v_, Feb 10 2004 *)

%t With[{nn=30},CoefficientList[Series[Exp[ArcSin[x]],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Feb 26 2013 *)

%t Table[FullSimplify[2^(n-2) * (Exp[Pi/2]-(-1)^n*Exp[-Pi/2]) * Gamma[(n-I)/2] * Gamma[(n+I)/2] / Pi], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 06 2014 *)

%o (Maxima) a(n):=(n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2))*(-1)^(k-j),j,1,k))*binomial(k+n-1,n-1),k,1,n-m))/(m-1)!,m,1,n); /* _Vladimir Kruchinin_, Sep 12 2010 */

%Y Bisections are expansions of sin(arcsinh(x)) and cos(arcsinh(x)).

%Y Bisections are A101927 and A101928.

%Y Cf. A002019.

%Y Cf. A166741, A166748. - _Jaume Oliver Lafont_, Oct 24 2009

%K nonn

%O 0,4

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E More terms from _Christian G. Bower_