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P(n+1,t) = (1-t)^(2n+1) Sum_{k>=1} k^(n+k) [t*exp(-t)]^k / k! for n>0; consequently, Sum(_{k>=1} (-1)^k k^(n+k) x^k/k!= [1+LW(x)]^(-(2n+1))P[n+1,-LW(x)] where LW(x) is the Lambert W-Function and P(n,t), for n > 0, are the row polynomials as given in Copeland's 2008 comment. (End)
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Ming-Jian Ding and Jiang Zeng, <a href="https://arxiv.org/abs/2307.00566">Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions</a>, arXiv:2307.00566 [math.CO], 2023.
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