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A220819
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Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].
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2
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0, 1, 4, 14, 48, 171, 614, 2270, 8518, 32567, 126168, 495079, 1962752, 7853581, 31672502, 128622480, 525523990, 2158818376, 8911039462, 36941520279, 153740822408, 642085403709, 2690217364606, 11304538078369, 47630350694248, 201181246749072, 851690546714230
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OFFSET
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1,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1542 (first 200 terms from Andrew Howroyd)
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.
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FORMULA
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a(n) = A083563(n) - 2*A001190(n). - Andrew Howroyd, Sep 23 2018
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MAPLE
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b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
end:
a:= n-> b(n, 2)-2*b(n, 1):
seq(a(n), n=1..40); # Alois P. Heinz, Sep 07 2019
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MATHEMATICA
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(* 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}];
(* 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;
a[n_] := b[n] - 2 c[n];
Array[a, 27] (* Jean-François Alcover, Sep 07 2019 *)
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CROSSREFS
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Column k=2 of A319541.
Cf. A001190, A083563.
Sequence in context: A007851 A014325 A047028 * A047138 A111299 A245824
Adjacent sequences: A220816 A220817 A220818 * A220820 A220821 A220822
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Dec 22 2012
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EXTENSIONS
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Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018
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STATUS
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approved
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