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A144180
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Number of ways of placing n labeled balls into n unlabeled (but 5-colored) boxes.
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16
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1, 5, 30, 205, 1555, 12880, 115155, 1101705, 11202680, 120415755, 1362057155, 16151603830, 200144023805, 2584429030505, 34691478901030, 483040313859705, 6963313750468055, 103747357497925880, 1595132080103893655
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OFFSET
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0,2
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COMMENTS
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The number of ways of putting n labeled balls into a set of bags and then putting the bags into 5 labeled boxes. - Peter Bala, Mar 23 2013
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LINKS
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FORMULA
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G.f.: A(x) satisfies 5*(x/(1-x))*A(x/(1-x)) = A(x)-1; five times the binomial transform equals this sequence shifted one place left.
E.g.f.: exp(5*(exp(x)-1)).
G.f.: (G(0) - 1)/(x-1)/5 where G(k) = 1 - 5/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) ~ n^n * exp(n/LambertW(n/5)-5-n) / (sqrt(1+LambertW(n/5)) * LambertW(n/5)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 5^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1,
(1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*5)
end:
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MATHEMATICA
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PROG
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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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