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Here is a relevant paper:
T. Worsch, 1994. "Lower and upper bounds for (sums of) binomial coefficients". http://digbib.ubka.uni-karlsruhe.de/volltexte/181894Lower and upper bounds for (sums of) binomial coefficients
T. Worsch. "Lower and upper bounds for (sums of) binomial coefficients". http://digbib.ubka.uni-karlsruhe.de/volltexte/181894
T. Worsch, 1994. Lower and upper bounds for (sums of) binomial coefficients