A012020 Expansion of e.g.f.: sin(sin(arctan(x))) (odd powers only).

1, -4, 76, -3256, 245008, -28441216, 4700478784, -1047088053376, 302112622479616, -109527844826856448, 48716214653800569856, -26075068739563056830464, 16529214537740143196901376

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

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]

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

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

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

Example

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

Mathematica

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

Prog

(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 */

Crossrefs

Sequence in context: A187542 A364111 A009631 * A012041 A024258 A012101

Adjacent sequences: A012017 A012018 A012019 * A012021 A012022 A012023

Keyword

sign

Author

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

Extensions

Definition corrected by Joerg Arndt, May 19 2011

Status

approved