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%I #11 Feb 06 2015 09:14:42
%S 1,1,5,16,49,142,415,1198,3473,10048,29118,84376,244747,710198,
%T 2062273,5991417,17416400,50652247,147384675,429043389,1249508946,
%U 3640449678,10610613551,30937605075,90237313082,263288153073,768449666116,2243530461066,6552016136666
%N Number of unlabeled rooted trees with 2n+1 nodes and maximal outdegree (branching factor) n.
%H Alois P. Heinz, <a href="/A244410/b244410.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = A244372(2n+1,n).
%F a(n) ~ c * d^n / sqrt(n), where d = 2.955765285651994974714817524... is the Otter's rooted tree constant (see A051491), and c = 2.806733... . - _Vaclav Kotesovec_, Jul 11 2014
%p b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
%p end:
%p a:= n-> `if`(n=0, 1, b(2*n$2, n$2)-b(2*n$2, n-1$2)):
%p seq(a(n), n=0..30);
%t b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := If[n == 0, 1, b[2*n, 2 n, n, n] - b[2*n, 2 n, n - 1, n - 1]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)
%Y Cf. A244372, A244407, A051491.
%K nonn
%O 0,3
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 27 2014
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