A318234
Number of inequivalent leaf-colorings of totally transitive rooted trees with n nodes.
Original entry on oeis.org
1, 1, 2, 5, 13, 34, 87
Offset: 1
Inequivalent representatives of the a(6) = 34 leaf-colorings:
(11(11)) (11111) (111(1)) (1(111)) (1(1)(1))
(11(12)) (11112) (111(2)) (1(112)) (1(1)(2))
(11(22)) (11122) (112(1)) (1(122)) (1(2)(2))
(11(23)) (11123) (112(2)) (1(123)) (1(2)(3))
(12(11)) (11223) (112(3)) (1(222))
(12(12)) (11234) (123(1)) (1(223))
(12(13)) (12345) (123(4)) (1(234))
(12(33))
(12(34))
A339648
Number of series reduced trees with n nodes and integer labeled leaves covering an initial interval of positive integers.
Original entry on oeis.org
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
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)).
-
\\ 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)))}
Comments