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Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).
5

%I #11 Nov 03 2019 01:42:52

%S 2,1,2,3,6,10,22,42,94,203,470,1082,2602,6270,15482,38525,97258,

%T 247448,635910,1645411,4289010,11245670,29656148,78595028,209273780,

%U 559574414,1502130920,4046853091,10939133170,29661655793

%N Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

%H Andrew Howroyd, <a href="/A122086/b122086.txt">Table of n, a(n) for n = 1..500</a>

%F For n even, a(n) = 2*A000055(n) - A000081(n/2), for n odd, a(n) = 2*A000055(n).

%F G.f.: 2*f(x) - f(x)^2 where f(x) is the g.f. of A000081. - _Andrew Howroyd_, Nov 02 2019

%o (PARI) \\ here TreeGf is A000081 as g.f.

%o TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}

%o seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2)} \\ _Andrew Howroyd_, Nov 02 2019

%Y Row sums of A122085.

%Y Antidiagonal sums of A329054.

%Y Same as A125702 except for n = 1.

%Y Cf. A000055, A000081.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Oct 19 2006

%E Edited by _Christian G. Bower_ and _Franklin T. Adams-Watters_, Jan 05 2007