We analyze properties of the 2-adic valuations of the Stirling numbers of the second kind.

We analyze properties of the 2-adic valuation of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence.

The independence polynomial of a (finite) graph is the generating function for the number of independent sets of each cardinality. Assuming that each possible edge of a complete graph of order n is independently operational with probability p, we consider the expected independence polynomial. We show here that for all fixed p∈(0,1), the expected independence polynomials of complete graphs have all real, simple roots.

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ab) preserves the value of an elliptic integral, and its iteration produces the classical arithmeticgeometric mean AGM(a, b). We present analogous transformations for rational functions integrated over the whole real line.