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%I #8 Dec 13 2020 17:26:57
%S 1,0,2,4,16,62,290,1496,8548,53278,359076,2597052,20034252,163996372,
%T 1418326160,12911494594,123317867572,1232219079760,12848961783474,
%U 139505358593240,1573914932077692,18418287165450500,223191801317514104,2796501582165674166,36179439053130339742
%N Number of series reduced trees with n nodes and integer labeled leaves covering an initial interval of positive integers.
%C Only leaves are labeled.
%H Andrew Howroyd, <a href="/A339648/b339648.txt">Table of n, a(n) for n = 1..200</a>
%e a(4) = 4: (111), (112), (122), (123).
%e a(5) = 16: (1111), (1112), (1122), (1123), (1222), (1223), (1233), (1234), (1(11)), (1(12)), (1(22)), (1(23)), (2(11)), (2(12)), (2(13)), (3(12)).
%o (PARI) \\ here R(n,k) gives number of colorings with k colors as vector.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o R(n,k)={my(v=vector(n)); v[1]=k; for(n=2, #v, v[n] = EulerT(concat(v[1..n-2], [0]))[n-1]); v}
%o seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}
%Y Cf. A001678 (uncolored), A318231 (inequivalent colorings).
%K nonn
%O 1,3
%A _Andrew Howroyd_, Dec 11 2020
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