This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144180 #41 Aug 06 2024 04:36:01
%S A144180 1,5,30,205,1555,12880,115155,1101705,11202680,120415755,1362057155,
%T A144180 16151603830,200144023805,2584429030505,34691478901030,
%U A144180 483040313859705,6963313750468055,103747357497925880,1595132080103893655
%N A144180 Number of ways of placing n labeled balls into n unlabeled (but 5-colored) boxes.
%C A144180 a(n) is also the exp transform of A010716. - _Alois P. Heinz_, Oct 09 2008
%C A144180 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
%H A144180 Alois P. Heinz, <a href="/A144180/b144180.txt">Table of n, a(n) for n = 0..200</a>
%H A144180 H. D. Nguyen, D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
%H A144180 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A144180 a(n) = Sum_{k=0..n} 5^k * A048993(n,k); A048993: Stirling2 numbers.
%F A144180 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.
%F A144180 E.g.f.: exp(5*(exp(x)-1)).
%F A144180 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
%F A144180 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
%F A144180 G.f.: Sum_{j>=0} 5^j*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 07 2019
%p A144180 a:= proc(n) option remember; `if`(n=0, 1,
%p A144180 (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*5)
%p A144180 end:
%p A144180 seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 09 2008
%t A144180 Table[BellB[n,5],{n,0,20}] (* _Vaclav Kotesovec_, Mar 12 2014 *)
%o A144180 (Sage) expnums(19, 5) # _Zerinvary Lajos_, May 15 2009
%Y A144180 Cf. A000110, A001861, A027710, A078944. A144223, A144263, A189233, A221159, A221176.
%K A144180 nonn
%O A144180 0,2
%A A144180 _Philippe Deléham_, Sep 12 2008
%E A144180 More terms from _Alois P. Heinz_, Oct 09 2008