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Notamathematician
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Closed form for the number of steps required to get $n$ balls in the last box
@PeterTaylor, could you please improve the comment in A077071?
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revised
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Closed form for the number of steps required to get $n$ balls in the last box
@PeterTaylor, thank you for comment! Please try to write a prog (for example on PARI) and check the result. If that does not help, just tell me and I will add mine prog which most likely will make things absolutely clear.
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Prime numbers and number of partitions of $n$ into distinct parts with boundary size $2$
@FedorPetrov, thank you for comment! Sorry, mistake was fixed. Try again now.
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Closed form for coefficients related to excedance set of permutation
@Wolfgang, thank you for comment! I am the author of this sequence. I also did a lot of work to find the simplest formula possible. I think the simplest is the variation of the Gandhi polynomials (see A357990). On the other hand, I also found a closed form alternative to the inverse binomial transformation modulo 2 (see draft for A358612). Can you suggest something even simpler?
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Closed form for coefficients related to excedance set of permutation
@MartinRubey, thank you for comment! Number of non-zero $T(n,k)$ for a given $n$ is $\operatorname{wt}(n)+2$ where $\operatorname{wt}(n)$ is the binary weight of $n$ (OEIS A000120). Also I will try to check your first comment later.
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Closed form for coefficients related to excedance set of permutation
@PeterTaylor, thank you for comment! Done (also sign was fixed). Please try to compute coefficients again.
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