A339648 Number of series reduced trees with n nodes and integer labeled leaves covering an initial interval of positive integers.

1, 0, 2, 4, 16, 62, 290, 1496, 8548, 53278, 359076, 2597052, 20034252, 163996372, 1418326160, 12911494594, 123317867572, 1232219079760, 12848961783474, 139505358593240, 1573914932077692, 18418287165450500, 223191801317514104, 2796501582165674166, 36179439053130339742

Offset: 1

Author

Andrew Howroyd, Dec 11 2020

Keywords

• nonn

Comments

Only leaves are labeled.

Examples

			a(4) = 4: (111), (112), (122), (123).
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)).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A001678 (uncolored), A318231 (inequivalent colorings).

Programs

PARI

\\ here R(n,k) gives number of colorings with k colors as vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
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}
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}