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How many ways to pick AT MOST k elements from n elements?
By tactical-teto, history, 20 months ago, In English

Given a fixed k (k<=1e6), make an array F of size n (n<=1e6) , with F[i] be the number of ways to pick AT MOST k elements from i elements.

Is there a way to create this array efficiently?

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$$$F_i$$$ is the number of ways to pick at most $$$k$$$ elements from $$$i$$$ elements total.

If you don't pick the first element, you have $$$F_{i-1}$$$ ways to continue.

If you pick the first element, you need to choose at most $$$k-1$$$ from the remaining $$$i-1$$$. This is $$$F_{i-1}$$$ minus the number of ways to pick exactly $$$k$$$ from the remaining $$$i-1$$$ elements, so $$$F_{i-1} - \binom{i-1}{k}$$$.

In total $$$F_{i} = 2F_{i-1} - \binom{i-1}{k}$$$, you can compute the binomials quickly by precomputing factorials.

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