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 A006228 Expansion of exp(arcsin(x)). (Formerly M1523) 15
 1, 1, 1, 2, 5, 20, 85, 520, 3145, 26000, 204425, 2132000, 20646925, 260104000, 2993804125, 44217680000, 589779412625, 9993195680000, 151573309044625, 2898026747200000, 49261325439503125, 1049085682486400000, 19753791501240753125, 463695871658988800000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 REFERENCES L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116. Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010. FORMULA 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 a(0)=1, a(1)=1, a(n) = (1+(n-2)^2)*a(n-2) for n >= 2. Jaume Oliver Lafont, Oct 24 2009 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 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 a(n) ~ (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 23 2013 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 MAPLE a:= n-> n!*coeff(series(exp(arcsin(x)), x, n+1), x, n): seq(a(n), n=0..25); # Alois P. Heinz, Aug 17 2018 MATHEMATICA Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (* Robert G. Wilson v, Feb 10 2004 *) With[{nn=30}, CoefficientList[Series[Exp[ArcSin[x]], {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Feb 26 2013 *) 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 *) PROG (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 */ CROSSREFS Bisections are expansions of sin(arcsinh(x)) and cos(arcsinh(x)). Bisections are A101927 and A101928. Cf. A002019. Cf. A166741, A166748. - Jaume Oliver Lafont, Oct 24 2009 Sequence in context: A192101 A012768 A170947 * A363140 A190656 A262166 Adjacent sequences: A006225 A006226 A006227 * A006229 A006230 A006231 KEYWORD nonn AUTHOR N. J. A. Sloane, Jeffrey Shallit EXTENSIONS More terms from Christian G. Bower STATUS approved

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Last modified December 11 02:45 EST 2023. Contains 367717 sequences. (Running on oeis4.)