I have implemented the following thing.
There are two arrays:
dp1[j] and dp2[j].
It contains count of such sequenses that they end at exactly j lemons.
dp1[0] is always 1.
dp1[1] is 1 iff K>0.
There is a loop from i=2 to N.
On the j-th iteration I calculate answer for length j.
But it does not works at all!!
And I do not have any clue why.
Help me understand why.

#include <stdio.h>

int main(){
    unsigned long long dp1[10001];
    unsigned long long dp2[10001];
    for(int j=1; j<=10000;++j)
         dp1[j]=0, dp2[j]=0;
    dp1[0]=1;
    dp2[0]=0;
    int N,K;
    scanf("%d %d",&N,&K);
   if(K>0) dp1[1]=1;
   for(int j=2;j<=N;++j){
         for(int k=1; k<=K && k<=j;++k)
              dp2[k]=dp1[k-1];
        for(int k=0; k<=K && k<=j-1;++k)
               dp2[0]+=dp1[k];
         for(int k=0; k<=K && k<=j;++k)
             dp1[k]=dp2[k];
    }
    unsigned long long ans=0;
    for(int k=0; k<=K ;++k)
        ans+=dp1[k];
    printf("%llu",ans);
    return 0;
}

It was because I didn't update dp2.

Very cute problem!
My code was TERRIBLE.
I completely rewrote it and got AC 0.2 seconds with Python3.