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COMMENTS
| Smallest integer value of the form 1/z(k,n) where z(k,x)=x/(x-1)^2-sum(i=1,k,i/x^i).
For any x>1 lim k -> infinity z(k,x)=0. More generally if p is an integer >=2, 1/z(u(k),p) is an integer for any k>=2 where u(k)=(p-1)^2*p^((p^k-(p-1)*k-p)/(p-1)). u(k) can also be written : u(k)=(p-1)^2*p^(1+p+p^2+...+p^(k-2))
For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,...,n} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan R. Janjic (agnus(AT)blic.net), May 10 2007
With offset = 1, a(n) = Sum_{k=0...n} binomial(n,k)*n^k*k which enumerates the total number of elements in the domain of definition over all partial functions on n labelled objects. - Geoffrey Critzer, Feb 08 2012
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MATHEMATICA
| Table[Sum[Binomial[n, k] n^k k, {k, 0, n}], {n, 1, 20}] (* Geoffrey Critzer, Feb 08 2012 *)
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