A012020 - OEIS

%I #19 Oct 21 2015 11:25:11

%S 1,-4,76,-3256,245008,-28441216,4700478784,-1047088053376,

%T 302112622479616,-109527844826856448,48716214653800569856,

%U -26075068739563056830464,16529214537740143196901376

%N Expansion of e.g.f.: sin(sin(arctan(x))) (odd powers only).

%F a(n) = (2*n+1)!*(-1)^n*sum(j=1..n+1, binomial((2*n-1)/2,(n+1-j))/(2*j-1)!). [_Vladimir Kruchinin_, May 19 2011]

%F a(n) = (2*n+1)! * [x^(2*n+1)] sin(sin(arctan(x))).

%F a(n) = -4*(3*n^2-3*n+1)*a(n-1) - 12*(n-1)^2*(2*n-3)*(2*n-1)*a(n-2) - 4*(n-2)*(n-1)*(2*n-5)*(2*n-3)^2*(2*n-1)*a(n-3). - _Vaclav Kotesovec_, Nov 08 2013

%F a(n) ~ (-1)^n * (2*n)^(2*n+2/3) * exp(3/2*2^(1/3)*n^(1/3)-2*n) / sqrt(3) * (1 - 19/72*2^(2/3)/n^(1/3) + 1849/5184*2^(1/3)/n^(2/3)). - _Vaclav Kotesovec_, Nov 08 2013

%e sin(sin(arctan(x)))=x-4/3!*x^3+76/5!*x^5-3256/7!*x^7+245008/9!*x^9-+...

%t Table[n!*SeriesCoefficient[Sin[x/Sqrt[1+x^2]],{x,0,n}],{n,1,41,2}] (* _Vaclav Kotesovec_, Nov 08 2013 *)

%o (Maxima) a(n):=(2*n+1)!*(-1)^n*sum(binomial((2*n-1)/2,(n+1-j))/(2*j-1)!,j,1,n+1); /* _Vladimir Kruchinin_, May 19 2011 */

%K sign

%O 0,2

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

%E Definition corrected by _Joerg Arndt_, May 19 2011