%I #17 Sep 07 2019 12:26:03

%S 0,1,4,14,48,171,614,2270,8518,32567,126168,495079,1962752,7853581,

%T 31672502,128622480,525523990,2158818376,8911039462,36941520279,

%U 153740822408,642085403709,2690217364606,11304538078369,47630350694248,201181246749072,851690546714230

%N Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

%H Alois P. Heinz, <a href="/A220819/b220819.txt">Table of n, a(n) for n = 1..1542</a> (first 200 terms from Andrew Howroyd)

%H V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.

%F a(n) = A083563(n) - 2*A001190(n). - _Andrew Howroyd_, Sep 23 2018

%p b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,

%p (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))

%p end:

%p a:= n-> b(n, 2)-2*b(n, 1):

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Sep 07 2019

%t (* b = A083563 *) b[n_] := b[n] = If[n < 2, 2*n, If[OddQ[n], 0, #*(1 - #)/2 &[b[n/2]]]] + Sum[b[i]*b[n - i], {i, 1, n/2}];

%t (* c = A001190 *) c[n_?OddQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, (n - 1)/2}]; c[n_?EvenQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, n/2 - 1}] + (1/2)*c[n/2]*(1 + c[n/2]); c[0] = 0; c[1] = 1;

%t a[n_] := b[n] - 2 c[n];

%t Array[a, 27] (* _Jean-François Alcover_, Sep 07 2019 *)

%Y Column k=2 of A319541.

%Y Cf. A001190, A083563.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Dec 22 2012

%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 23 2018