For me the result in the end is $f(1,\pi)=\frac{1}{3!}(-e^{\pi}-3\pi e^{\pi})(x-1)(x-\pi)^3$
I think there's a missing $\frac{1}{n!}$ in your formula, and I think your derivatives aren't correct. At any rate, in Mathematica you can just use SeriesCoefficient:
SeriesCoefficient
SeriesCoefficient[Exp[x y] Sin[x^2 y], {x, 1, 1}, {y, Pi, 3}]
1/3 E^? (-3 + ?)
