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  • Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor. $$
  • Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$) such that $$ T(n, k) = \begin{cases} 1 & \textrm{if } n = 0, k = 1 \\ 1 & \textrm{if } n = 1, k = 1 \\ 1 & \textrm{if } n = 1, k = 2 \\ T(n-2^{\ell(n)-1},k) & \textrm{if } n < 3\cdot2^{\ell(n)-1}, 1 \leqslant k < f(n) \\ T(n-2^{\ell(n)-1},f(n)-1) & \textrm{if } n < 3\cdot2^{\ell(n)-1}, k = f(n) \\ \sum\limits_{i=1}^{k} iT(n-2^{\ell(n)},k) & \textrm{otherwise} \end{cases} $$
  • Let $a(n)$ be an integer sequence such that $$ a(n) = T(n, f(n)). $$
  • Let $s(n)$ be an integer sequence such that $$ s(n) = \sum\limits_{i=0}^{2^n-1} a(i). $$

I conjecture that $s(n)$ is somehow related with Laguerre polynomials.

Is there a way to get a nice closed form or generating function for $s(n)$?

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    $\begingroup$ "somehow related". been slacking? :P $\endgroup$
    – mathworker21
    Commented 19 hours ago
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    $\begingroup$ I continue to work on it, but there is no result yet. $\endgroup$
    – Notamathematician
    Commented 19 hours ago

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