Integral representations and properties of several finite sums containing central binomial coefficients

For m > 0 and n is an element of N, let S-m,n(+/-) = Sigma(n)(k=1)((2k)(k)) (1/2(2k))k (+/- 1)(k)/k+m . In the paper, the authors establish the integral k=1 k22k Pn representations, positivity, monotonic properties, convex properties, and limits of the sequences S-m,n(+/-) and Sigma(n)(k=0)((2k)(k)) (+/- 1)(k)/2(2k) for n is an element of N. Several of these results recover the corresponding known conclusions.