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A220821 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [4].

%I A220821 #15 Sep 07 2019 12:31:25
%S A220821 0,0,0,15,240,2604,24180,207732,1710108,13739550,108853512,855732465,
%T A220821 6700902804,52395480996,409733313444,3207687963129,25155951725808,
%U A220821 197703130100532,1557413160706764,12298597436673711,97359729090421320,772615510913274126,6145842794363133324
%N A220821 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [4].
%H A220821 Andrew Howroyd, <a href="/A220821/b220821.txt">Table of n, a(n) for n = 1..200</a>
%H A220821 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.
%p A220821 b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
%p A220821       (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
%p A220821     end:
%p A220821 a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(4):
%p A220821 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 07 2019
%t A220821 A[n_, k_] := A[n, k] = If[n < 2, k n, If[OddQ[n], 0, (# (1 - #)/2)&[A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]];
%t A220821 T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];
%t A220821 a[n_] := T[n, 4];
%t A220821 Array[a, 23] (* _Jean-François Alcover_, Sep 02 2019, after _Alois P. Heinz_ in A319541 *)
%Y A220821 Column k=4 of A319541.
%K A220821 nonn
%O A220821 1,4
%A A220821 _N. J. A. Sloane_, Dec 22 2012
%E A220821 Terms a(11) and beyond from _Andrew Howroyd_, Sep 23 2018