# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a006228 Showing 1-1 of 1 %I A006228 M1523 #44 Aug 17 2018 13:41:40 %S A006228 1,1,1,2,5,20,85,520,3145,26000,204425,2132000,20646925,260104000, %T A006228 2993804125,44217680000,589779412625,9993195680000,151573309044625, %U A006228 2898026747200000,49261325439503125,1049085682486400000,19753791501240753125,463695871658988800000 %N A006228 Expansion of exp(arcsin(x)). %D A006228 L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150. %D A006228 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A006228 T. D. Noe, Table of n, a(n) for n = 0..100 %H A006228 H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116. %H A006228 Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010. %F A006228 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 A006228 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 A006228 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 A006228 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 A006228 a(n) ~ (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Oct 23 2013 %F A006228 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 A006228 a:= n-> n!*coeff(series(exp(arcsin(x)), x, n+1), x, n): %p A006228 seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 17 2018 %t A006228 Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (* _Robert G. Wilson v_, Feb 10 2004 *) %t A006228 With[{nn=30},CoefficientList[Series[Exp[ArcSin[x]],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Feb 26 2013 *) %t A006228 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 A006228 (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 A006228 Bisections are expansions of sin(arcsinh(x)) and cos(arcsinh(x)). %Y A006228 Bisections are A101927 and A101928. %Y A006228 Cf. A002019. %Y A006228 Cf. A166741, A166748. - _Jaume Oliver Lafont_, Oct 24 2009 %K A006228 nonn %O A006228 0,4 %A A006228 _N. J. A. Sloane_, _Jeffrey Shallit_ %E A006228 More terms from _Christian G. Bower_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE