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Fedor Petrov
Fedor Petrov
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The generating function is $$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^k},$$$$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^n},$$ so it is the same as the number of partitions onto summands not belonging to the set $\{k^2+k:k=1,2,\dots\}$.

The generating function is $$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^k},$$ so it is the same as the number of partitions onto summands not belonging to the set $\{k^2+k:k=1,2,\dots\}$.

The generating function is $$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^n},$$ so it is the same as the number of partitions onto summands not belonging to the set $\{k^2+k:k=1,2,\dots\}$.

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Fedor Petrov
Fedor Petrov
  • 108.9k
  • 9
  • 264
  • 459

The generating function is $$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^k},$$ so it is the same as the number of partitions onto summands not belonging to the set $\{k^2+k:k=1,2,\dots\}$.