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Number of (unlabeled) rooted trees with n leaf nodes and without unary nodes such that the maximum of the node outdegrees equals four.
2

%I #7 Sep 12 2017 11:54:04

%S 1,2,7,22,72,230,751,2442,8006,26280,86604,285994,946866,3140812,

%T 10438300,34747649,115849084,386779317,1292998720,4327654320,

%U 14500841169,48639319376,163308287353,548820437392,1845999502151,6214297279692,20935992503127,70586182742450

%N Number of (unlabeled) rooted trees with n leaf nodes and without unary nodes such that the maximum of the node outdegrees equals four.

%H Alois P. Heinz, <a href="/A292230/b292230.txt">Table of n, a(n) for n = 4..1000</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%e : a(6) = 7:

%e : o o o o

%e : / \ / \ /( )\ / | \

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

%e : / \ /( )\ / \ /( )\

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

%e : /( )\ ( ) ( )

%e : N N N N N N N N

%e :

%e : o o o

%e : / \ /( )\ / ( \ \

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

%e : /( )\ ( ) /|\ ( ) ( )

%e : N N N N N N N N N N N N N

%e :

%p b:= proc(n, i, v, k) option remember; `if`(n=0,

%p `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,

%p `if`(v=n, 1, add(binomial(A(i, k)+j-1, j)*

%p b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))

%p end:

%p A:= proc(n, k) option remember; `if`(n<2, n,

%p add(b(n, n+1-j, j, k), j=2..min(n, k)))

%p end:

%p a:= n-> A(n, 4)-A(n, 3):

%p seq(a(n), n=4..35);

%Y Column k=4 of A292086.

%K nonn

%O 4,2

%A _Alois P. Heinz_, Sep 12 2017