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In the paper, with the aid of the Fa\`a di Bruno formula, be virtue ofseveral identities for the Bell polynomials of the second kind, with the helpof two combinatorial identities, by means of the (logarithmically) completemonotonicity of generating functions of several integer sequences, and in thelight of Wronski's theorem, the authors establish Taylor's series expansions ofseveral functions involving the inverse (hyperbolic) tangent function, find outMaclaurin's series expansion of a complex function composited by inversetangent, square root, and exponential functions and posed by Herbert S. Wilf,and analyze some properties, including generating functions, limits,positivity, monotonicity, and logarithmic convexity, of the coefficients inMaclaurin's series expansion of Wilf's function. These coefficients inMaclaurin's series expansion of Wilf's function are closed-form expressions interms of the Stirling numbers of the second kind. The authors also derive aclosed-form formula for a sequence of special values of Gauss' hypergeometricfunction, discover a closed-form formula for a sequence of special values ofthe Bell polynomials of the second kind, present several infinite seriesrepresentations of the circular constant Pi and other sequences, recover anasymptotic rational approximation to the circular constant Pi, and connectseveral integer sequences by determinants.

Closed-form formulas and properties of coefficients in Maclaurin's  series expansion of Wilf's function composited by inverse tangent, square  root, and exponential functions