This paper explores sums of the type \(\sum_{0\leq k\leq n}\left( -1\right) ^{k}k^{m}\binom{n}{k}^{-1}\) supplying recurrence formulae, generating functions and closed formulae in terms of the matrix introduced by S. Akiyama and Y. Tanigawa [Ramanujan J. 5, No. 4, 327–351 (2001; Zbl 1002.11069)], defined by D. Merlini et al. [Integers 5, No. 1, Paper A05, 12 p. (2005; Zbl 1087.11012)] and, differently, by Y. Inaba [J. Integer Seq. 8, No. 2, Art. 05.2.7, 6 p. (2005; Zbl 1073.11009)], applied by K.-W. Chen [J. Integer Seq. 4, No. 1, Art. 01.1.6, 7 p. (2001; Zbl 0973.11021)] and by M. Kaneko [J. Integer Seq. 3, No. 2, Art. 00.2.9 (2000; Zbl 0982.11009)].
Beyond the Bernoulli, Worpitzky, Stirling and Eulerian numbers (all illustrated, e.g., by M. Abramowitz (ed.) and I. A. Stegun (ed.) [Handbook of mathematical functions with formulas, graphs and mathematical tables. Washington: U.S. Department of Commerce (1964; Zbl 0171.38503)], by R. L. Graham et al. [Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley (1994; Zbl 0836.00001)] and in the OEIS by N. J. A. Sloane [Ann. Math. Inform. 41, 219–234 (2013; Zbl 1274.11001)]), in the proof the authors employ an observation given by R. Sprugnoli [J. Integer Seq. 14, No. 7, Article 11.7.8, 14 p. (2011; Zbl 1275.60016)], an identity supplied by B. Sury et al. [J. Integer Seq. 7, No. 2, Art. 04.2.8, 12 p. (2004; Zbl 1069.11008)] and a theorem found by T. Mansour [Adv. Appl. Math. 28, No. 2, 196–202 (2002; Zbl 1004.05011)] while generalizing the idea of B. Sury [Eur. J. Comb. 14, No. 4, 351–353 (1993; Zbl 0783.05002)] to connect the inverse of the binomial coefficients to the Beta function.
Many papers on sums involving inverses of binomial coefficients are recalled, both by the same H. Belbachir et al. [J. Integer Seq. 14, No. 6, Article 11.6.6, 16 p. (2011; Zbl 1232.11023)] and by other authors, e.g., J. Pla [Fibonacci Q. 35, No. 4, 342–345 (1997; Zbl 0915.11013)], A. M. Rockett [Fibonacci Q. 19, 434–437 (1981; Zbl 0476.05011)], A. Sofo [J. Integer Seq. 9, No. 4, Article 06.4.5, 13 p. (2006; Zbl 1108.11021)], R. Sprugnoli [Integers 6, Paper A27, 18 p. (2006; Zbl 1106.05003)], T. B. Staver [Norsk Mat. Tidsskr. 29, 97–103 (1947; Zbl 0030.28901)], J.-H. Yang and F.-Z. Zhao [J. Integer Seq. 9, No. 4, Article 06.4.2, 11 p. (2006; Zbl 1108.11022)], F.-Z. Zhao and T. Wang [Integers 5, No. 1, Paper A22, 5 p. (2005; Zbl 1085.11008)], T. Trif [Fibonacci Q. 38, No. 1, 79–84 (2000; Zbl 0943.11014)] and, in particular, K. N. Boyadzhiev [Fibonacci Q. 46–47, No. 4, 326–330 (2009; Zbl 1232.11032)] who provided the alternative proof of a lemma here necessary to express the alternating sums of the reciprocals of binomial coefficients via the Akiyama-Tanigawa matrix.