A remark on the Boros-Moll sequence (Q2882730)

In this paper the author improves the results of \textit{H. Prodinger} [Integers 11, No. 2, 163--168 (2011; Zbl 1237.11005)] on the asymptotic behavior of the function NEWLINE\[NEWLINEf_l (m) = \nu_2 (A_{l,l+(m-1)2^{\nu_2(l)+1}}),NEWLINE\]NEWLINE where \(\nu_2 (n)\) denotes the exponent of the highest power of \(2\) dividing \(n\), and NEWLINE\[NEWLINE A_{l,m} = l!\, m!\,2^{-(m-l)} \sum_{1 \leq k \leq m} 2^k {{2m-2k} \choose {m-k}} {{m+k} \choose m}{k \choose l};NEWLINE\]NEWLINE this sum arises in the evaluation of a certain integral. Prodinger found interesting expressions for \(\sum_{1 \leq k < n} f_l (k) (n-k)\) for \(l = 3,5\). Using results of \textit{H. Delange} [Enseign. Math., II. Sér. 21, 31--47 (1975; Zbl 0306.10005)], the author extends this to all odd \(l \geq 3\).