Let $k,n\in\mathbb{N}$. Let $N(k,n)$ denote the size of the finite set $$\{(x_1,\cdots,x_k)\in\mathbb{N}^k:x_1+2x_2+\cdots+kx_k=n\}. $$
I feel it special and important. Do $N(k,n)$ have names? Is there a table of values?
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1Related: https://math.stackexchange.com/questions/2053319/number-of-solutions-of-x-12x-2-cdotskx-k-n?rq=1 – JMoravitz Mar 28 '24 at 14:48
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1As for a name and easy closed form, neither exist to my knowledge. Approximations exist, read the linked post for one such method to approximate. – JMoravitz Mar 28 '24 at 14:50
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This is the number of partitions of $n-k$ into parts up to $k$. – joriki Mar 28 '24 at 16:28
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While not a name, it's certainly related to Faà di Bruno's formula. – Greg Martin Mar 28 '24 at 16:39
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1$N(k,n)$ is in bijection with the set of integer partitions of $n$ where every part has size at most $k$. In this correspondence, $x_i$ represents the number of parts with size $i$. It is also in bijection with partitions of $n$ with at most $k$ parts (via conjugation). Therefore, your question is a duplicate of this one: Number of ways to write $n$ as a sum of $k$ nonnegative integers. – Mike Earnest Mar 29 '24 at 00:05