A012065 Expansion of e.g.f: tan(arcsin(arcsin(x))).

1, 4, 84, 4152, 370128, 51861888, 10494283968, 2894815734912, 1043916757274880, 476720372375608320, 268870416029396075520, 183537887154798761809920, 149132786692921038502318080

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..215

Formula

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

E.g.f.: arcsin(x) / sqrt(1-arcsin(x)^2). - Vaclav Kotesovec, Feb 06 2015

a(n) ~ (2*n+1)! * sqrt(cos(1)) / (sqrt(Pi*n) * (sin(1))^(2*n+3/2)). - Vaclav Kotesovec, Feb 06 2015

Example

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

Mathematica

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 *)

Prog

(Maxima)
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 */
(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

Crossrefs

Sequence in context: A361542 A024260 A099706 * A012139 A012037 A012025

Adjacent sequences: A012062 A012063 A012064 * A012066 A012067 A012068

Keyword

nonn

Author

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

Status

approved