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A095176
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E.g.f.: exp(9x)/(1-9x)^(1/9).
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0
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1, 10, 109, 1432, 26497, 754894, 30787885, 1603546156, 99602138593, 7128277455538, 576063289419661, 51832424202980320, 5136461847251936929, 555721381650431686582, 65167921144448534609677
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OFFSET
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0,2
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COMMENTS
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Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n), A094935(n) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
In general, for k > 0, if e.g.f. = exp(k*x) / (1 - k*x)^(1/k), then a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * k^n / (Gamma(1/k) * exp(n-1) * n^(1 - 1/k)).
Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n} A046716(n, k)*9^k.
D-finite with recurrence a(n) +(-9*n-1)*a(n-1) +81*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 9^n / (Gamma(1/9) * exp(n-1) * n^(8/9)). - Vaclav Kotesovec, Nov 19 2021
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[Exp[9x]/Surd[1-9x, 9], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jul 25 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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