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MATHEMATICA
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nmax = 18; CoefficientList[Series[Sum[(Exp[x] - 1)^j/Product[(1 - k (Exp[x] - 1)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[Sum[j! BellB[j] x^j/Product[(1 - k x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[StirlingS2[n, k] k! BellB[k], {k, 0, n}], {n, 0, 18}]
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