Starling Sum

Let \(A(n,k)=\sum_{S\in[n]^{k}}\frac{1}{\Pi(S)}\).

Then \(A(200,197)\) can be expressed as \(x\times10^{-363}\). Find \(x\) up to 3 decimal places.

Details and Notations

i. \([n]\) means the set \(\{1,2,...,n\}.\)

ii. \([n]^{k}\) means the set of all subsets of \([n]\) that have \(k\) elements.

iii. \(\in\) means belongs to, i.e, \(S\in[n]^{k}\) means \(S\) is a subset of \([n]\) with \(k\) elements.

iv. \(\Pi(S)\) denotes the product of the elements of \(S\).

v. The sum is over all \(k\)-element subsets \(S\) of \([n]\).

vi. By convection \(\Pi(\phi)=1\) where \(\phi\) denotes the empty set.

vii. One must know about Stirling Numbers in order to solve it.

Example

\(A(n,1)=\sum_{S∈[n]^{1}}\frac{1}{\Pi(S)}=1+\frac{1}{2}+\cdots+\frac{1}{n}\)

\(A(n,2)=\sum_{S∈[n]^{2}}\frac{1}{\Pi(S)}=1+\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{2\times4}+\cdots\)

Try other interesting combinatorics in my set Hard

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