The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A012065 Expansion of e.g.f: tan(arcsin(arcsin(x))). 1

%I

%S 1,4,84,4152,370128,51861888,10494283968,2894815734912,

%T 1043916757274880,476720372375608320,268870416029396075520,

%U 183537887154798761809920,149132786692921038502318080

%N Expansion of e.g.f: tan(arcsin(arcsin(x))).

%H G. C. Greubel, <a href="/A012065/b012065.txt">Table of n, a(n) for n = 0..215</a>

%F a(n) = ((2*n+1)!*Sum_{m=0..n} binomial(m-1/2,m)*(2*m+1)!*(Sum_{j=0..n-m} (-1)^(j)*(Sum_{i=0..2*j} (2^i*stirling1(2*m+1+i,2*m+1) *binomial(2*j+2*m,2*m+i))/(2*m+1+i)!))*binomial(n-1/2,n-j-m))))). - _Vladimir Kruchinin_, Jun 15 2011

%F E.g.f.: arcsin(x) / sqrt(1-arcsin(x)^2). - _Vaclav Kotesovec_, Feb 06 2015

%F a(n) ~ (2*n+1)! * sqrt(cos(1)) / (sqrt(Pi*n) * (sin(1))^(2*n+3/2)). - _Vaclav Kotesovec_, Feb 06 2015

%e tan(arcsin(arcsin(x))) = x + 4/3!*x^3 + 84/5!*x^5 + 4152/7!*x^7 + 370128/9!*x^9 ...

%t With[{nn=30},Take[CoefficientList[Series[Tan[ArcSin[ArcSin[x]]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Oct 11 2014 *)

%o (Maxima)

%o a(n):=((2*n+1)!*sum(binomial(m-1/2,m)*(2*m+1)!*(sum((-1)^(j)*(sum((2^i*stirling1(2*m+1+i,2*m+1)*binomial(2*j+2*m,2*m+i))/(2*m+1+i)!,i,0,2*j))*binomial(n-1/2,n-j-m),j,0,n-m)),m,0,n)); /* _Vladimir Kruchinin_, Jun 15 2011 */

%o (PARI) x='x+O('x^50); v=Vec(serlaplace(asin(x) / sqrt(1-asin(x)^2))); vector(#v\2, n, v[2*n-1]) \\ _G. C. Greubel_, Apr 11 2017

%K nonn

%O 0,2

%A Patrick Demichel (patrick.demichel(AT)hp.com)

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 24 19:07 EDT 2023. Contains 361510 sequences. (Running on oeis4.)