%I #9 Jan 09 2020 19:22:52

%S 1,3,18,194,2944,57959,1398858,39981994,1320143478,49439258516,

%T 2070409961552,95867076538834,4863079990663528,268198764863998103,

%U 15977057268090388836,1022415045656417706598,69946606996018140613292,5094427098628436561252367,393558075509405403487404506

%N Number of series-reduced rooted trees whose leaves are sets with a total of n elements covering an initial interval of positive integers.

%H Andrew Howroyd, <a href="/A330764/b330764.txt">Table of n, a(n) for n = 1..200</a>

%e The a(3) = 18 trees:

%e (123) ((1)(12)) ((1)(1)(1))

%e ((1)(23)) ((2)(12)) ((1)((1)(1)))

%e ((2)(13)) ((1)(2)(2))

%e ((3)(12)) ((1)(1)(2))

%e ((1)(2)(3)) ((1)((2)(2)))

%e ((1)((2)(3))) ((1)((1)(2)))

%e ((2)((1)(3))) ((2)((1)(2)))

%e ((3)((1)(2))) ((2)((1)(1)))

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(k, n)]))[n])); v}

%o seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}

%Y Row sums of A330763.

%Y Cf. A330469 (leaves are multisets).

%K nonn

%O 1,2

%A _Andrew Howroyd_, Dec 29 2019