I don't have the answer, but to provide some data, here is the expansion of the derivatives of exp(sin(0)) for n=0,1,2,...,15:
1, 1, 1, 0, -3, -8, -3, 56, 217, 64, -2951, -12672, 5973, 309376, 1237173, -2917888.
The OEIS gives <a href="http://www.research.att.com/projects/OEIS?Anum=A002017">A002017</a>, which doesn't provide a general formula unfortunately.
If, in f<sub>n</sub>(t), we let s=sin(t) and c=cos(t), then the expansion looks like this:
f<sub>0</sub>(t) = 1s<sup>0</sup>c<sup>0</sup>.
f<sub>1</sub>(t) = 1s<sup>0</sup>c<sup>1</sup>.
f<sub>2</sub>(t) = -1s<sup>1</sup>c<sup>0</sup> + 1s<sup>0</sup>c<sup>2</sup>.
f<sub>3</sub>(t) = -1s<sup>0</sup>c<sup>1</sup> - 3s<sup>1</sup>c<sup>1</sup> + 1s<sup>0</sup>c<sup>3</sup>.
And so on. The coefficients here are trinomials. If I get any further I'll post here.