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for questions involving inequalities, upper and lower bounds.
2
votes
Sum over integer compositions
Your case p=1 seems to be http://oeis.org/A078812, just binomial(n+k-1, 2*k-1). See also the references to http://oeis.org/A078812.
The case p=-1 seems to be a variant of http://oeis.org/A048594, more …