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If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then $$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!}x^2+\frac{a_1^3+3a_1a_2+a_3}{3!}x^3+\cdots$$ is related to the number of set partitions. A particular case $f(x)=e^x-1$ results in the Bell numbers via $e^{e^x-1}$.

Consider now the choice of $f(x)=\sin x$. I would like to ask:

QUESTION. Is it true that $b_n=0$ iff $n=3$ when $e^{\sin x}=\sum_{n\geq0}\frac{b_n}{n!}x^n$?

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    $\begingroup$ This sequence is oeis.org/A002017. Perhaps this would help someone to answer $\endgroup$
    – Arsenii Sagdeev
    Commented Jul 11, 2022 at 20:21
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    $\begingroup$ $b_m=\sum_{k=0}^{\lfloor \frac{m}{2}\rfloor} (-1)^k \sigma_m(m-2k)$, where $\sigma_m(n)$ is number of ways to put $m$ distinct objects in $n$ identical groups each having odd number of elements. $\endgroup$
    – Alapan Das
    Commented Jul 12, 2022 at 3:13

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