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A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!. 2
1, 1, 13, 501, 38617, 4945385, 944469221, 250727790173, 88106527550129, 39555449833828817, 22093952731139969213, 15041143328788464370373, 12273562321018687866908553, 11833097802606125967312406457 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It appears that there are no negative terms.

It also appears that, if asin(x) is changed to asinh(x) in the definition, the sequence obtained is the same except alternating in sign: 1, -1, 13, -501, ... - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Jul 16 2009

LINKS

Table of n, a(n) for n=0..13.

FORMULA

a(n)=T(2*n+1,1)*2^n*(2*n+1)!, T(n,m)=if n=m then 1 else 1/2(Co(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), Co(n,m)=T121408(n,m)=(m!*(sum(k=0..n-m, (-1)^((k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((n-2)/2,(n-m-k)/2)))*((-1)^(n-m)+1))/2. [From Vladimir Kruchinin, Nov 11 2011]

EXAMPLE

F(x) =

x+1/2*x^3/3!+13/2^2*x^5/5!+501/2^3*x^7/7!+38617/2^4*x^9/9!+...

Special values:

F(x)=Pi/6 at x=F(1/2) = 0.51137532057552418592144885355...

F(x)=Pi/4 at x=F(sqrt(2)/2) = 0.74287348600976...

PROG

(PARI) {a(n)=local(A, B, F); F=asin(x+x*O(x^(2*n+1))); A=F; for(i=0, n, B=serreverse(A); A=(A+subst(B, x, F))/2); 2^n*(2*n+1)!*polcoeff(A, 2 *n+1, x)}

CROSSREFS

Cf. A095882, A095884, A095885.

Sequence in context: A281181 A183479 A203586 * A139168 A174564 A281055

Adjacent sequences:  A095880 A095881 A095882 * A095884 A095885 A095886

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 10 2004

STATUS

approved

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Last modified March 25 05:37 EDT 2017. Contains 284036 sequences.