Wolfram Language (Mathematica), 56 49 46 43 bytes

f[n_,a_]=If[n<=0,1+Sign@n,Tr[f[n-#,a]&/@a]] 

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Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

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Wolfram Language (Mathematica), 56 49 46 43 41 bytes

2 bytes golfed thanks to J42161217

n_~f~a_=If[n<1,1+Sign@n,Tr[f[n-#,a]&/@a]] 

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Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

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Wolfram Language (Mathematica), 56 49 46 43 bytes

f[n_,a_]=If[n<=0,1+Sign@n,Tr[f[n-#,a]&/@a]] 

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This borrows the method used by Arnauld's answer.

Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

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Wolfram Language (Mathematica), 56 49 46 43 bytes

f[n_,a_]=If[n<=0,1+Sign@n,Tr[f[n-#,a]&/@a]] 

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Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

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Wolfram Language (Mathematica), 56 49 46 bytes

f[n_,a_]=If[n<=0,Boole[n==0],Tr[f[n-#,a]&/@a]] 

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This borrows the method used by Arnauld's answer.

Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

Try it online!

Wolfram Language (Mathematica), 56 49 46 43 bytes

f[n_,a_]=If[n<=0,1+Sign@n,Tr[f[n-#,a]&/@a]] 

Try it online!

This borrows the method used by Arnauld's answer.

Initial solution:

Length[Join@@Permutations/@IntegerPartitions[#,∞,#2]]& 

Try it online!