login
Number of rooted trees with n nodes having some subtrees replaced by cycles.
3

%I #15 Jun 26 2022 02:20:19

%S 0,0,0,1,2,4,9,23,61,168,469,1326,3776,10833,31228,90438,262860,

%T 766497,2241194,6569206,19296214,56789286,167419568,494337282,

%U 1461690270,4327638394,12828158828,38067670764,113081627856,336233591365,1000636296475,2980391776958

%N Number of rooted trees with n nodes having some subtrees replaced by cycles.

%H Alois P. Heinz, <a href="/A213683/b213683.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A213674(n) - A000081(n).

%e : o : o o : o o o o :

%e : / \ : / \ | : / \ | / \ | :

%e : o---o : o o o : o o o o o o :

%e : : \ / / \ : | | / \ / \ | :

%e : : o o---o : o---o o o o---o o :

%e : : : \ / / \ :

%e : n=3 . n=4 : n=5 o o---o :

%e ...................................................................

%e : o o o o o o o o o :

%e : / \ | | | | / \ / \ / \ /|\ :

%e : o o o o o o o o o o o o o o o :

%e : | | / \ | / \ | / \ | | / \ / \ :

%e : o o o o o o o o o---o o o o o o---o :

%e : \ / | | / \ / \ | / \ \ / :

%e : o o---o o o o---o o o---o o :

%e : \ / / \ :

%e : n=6 o o---o :

%e :.................................................................:

%p b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0$2],

%p add(((x, y)-> map(p->binomial(p[1]+j-1, j)*p[2], [[x[1], y[1]],

%p [x[2], y[2]]]))(g(i), b(n-i*j, i-1)), j=0..n/i)))

%p end:

%p g:= n-> (l-> l+ [0, `if`(n>2, 1, 0)])(b(n-1, n-1)):

%p a:= n-> (l->l[2]-l[1])(g(n)):

%p seq(a(n), n=0..40);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A213674[i] + j - 1, j]*b[n - i*j, i - 1], {j, 0, n/i}]] // FullSimplify];

%t A213674[n_] := b[n - 1, n - 1] + If[n > 2, 1, 0];

%t A81[n_] := A81[n] = If[n <= 1, n, Sum[Sum[d*A81[d], {d, Divisors[j]}]*A81[n - j], {j, 1, n - 1}]/(n - 1)];

%t a[n_] := A213674[n] - A81[n];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 26 2022, after _Alois P. Heinz_ in A213674 *)

%Y Cf. A000081, A213674, A213682.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Mar 04 2013