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)).
Cf. A002019.
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Christian G. Bower
STATUS
approved