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A352593
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Denominator values occurring in formulas for the n-th integration of the Lambert W function.
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1
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1, 8, 648, 82944, 1296000000, 69984000000, 403443833184000000, 26440095051546624000000, 42154051662866968215552000000, 263462822892918551347200000000000, 826859199578154686310659783668531200000000000
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OFFSET
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1,2
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LINKS
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EXAMPLE
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If we use J(n, f(x)) notation for the n-th integration, then we can find the denominator
J(1, W(x)) = x/(W(x) (W(x)^2 - W(x) + 1) + c
J(2, W(x)) = x^2/(8W(x)^2) (4W(x)^3 - 6W(x)^2 + 6W(x) + 1) + kx + c
J(3, W(x)) = x^3/(648W(x)^3) (108W(x)^4 - 198W(x)^3 + 198W(x)^2 + 57W(x) + 8) + x^2/2 h + kx + c
...
where c, k, h are constants.
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MATHEMATICA
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max = 10; Table[Denominator[Together[Rest[NestList[Integrate[#, x] &, LambertW[x], max]]]][[k]] / ProductLog[x]^k, {k, 1, max}] (* Vaclav Kotesovec, Apr 14 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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