D[ProductLog[k, z], {z, n}] == Boole[n == 0, ProductLog[k, z]] + ((n! ProductLog[k, z]^n)/(z^n (1 + ProductLog[k, z])^(2 n - 1))) Sum[(((-1)^(n + q - j - 1) (l + n)^(j + l + n - 1) Binomial[l + n, n] Pochhammer[-2 n, q - j - l - m])/(j! (l + n)! (q - j - l - m)!)) ProductLog[k, z]^q, {q, 0, n}, {m, 0, q}, {l, 0, q - m}, {j, 0, q - l - m}] /; Element[n, Integers] && n > 0

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Eric Weisstein and Oleg Marichev

2007-05-02