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A290158
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a(n) = n! * [x^n] exp(-n*x)/(1 + LambertW(-x)).
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5
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1, 0, 4, -9, 208, -1525, 33516, -463099, 11293248, -231839577, 6517863100, -175791146311, 5723314711632, -189288946716181, 7083626583237036, -275649085963046475, 11724766124450058496, -522717581675749841713, 24981438186138642481404
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OFFSET
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0,3
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COMMENTS
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The n-th term of the n-th inverse binomial transform of A000312.
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LINKS
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FORMULA
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a(n) = (-1)^n * n! * [x^n] exp(n * x * (exp(x) - 1)).
a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} n^k * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} (k*x)^k / (1 + n*x)^(k+1).
a(n) = Sum_{k=0..n} (-n)^(n-k) * k^k * binomial(n,k). (End)
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MATHEMATICA
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Table[n! SeriesCoefficient[Exp[-n x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 18}]
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PROG
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(PARI) a(n) = (-1)^n*n!*sum(k=0, n\2, n^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, May 05 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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