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A213674
Number of rooted trees with n nodes, where cycles are allowed instead of subtrees.
3
0, 1, 1, 3, 6, 13, 29, 71, 176, 454, 1188, 3168, 8542, 23319, 64201, 178249, 498241, 1401344, 3962353, 11257882, 32122442, 92011118, 264474749, 762620137, 2205415254, 6394813039, 18587795338, 54151405093, 158088694125, 462420145673, 1355063144072
OFFSET
0,4
LINKS
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 3.0842695283018951389734653060490863..., c = 0.46707331868314508788370370390913... . - Vaclav Kotesovec, Sep 07 2014
EXAMPLE
: o : o : o o o : o o o o o o :
: : | : / \ / \ | : / \ | | / \ /|\ | :
: : o : o---o o o o : o o o o o o o o o o :
: : : | : \ / / \ / \ | | :
: : : o : o o---o o o o o :
: : : : | :
:n=1.n=2. n=3 . n=4 o :
...........................................................
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n-1, n-1) +`if`(n>2, 1, 0):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]] // FullSimplify] ; a[n_] := b[n-1, n-1] + If[n>2, 1, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Mar 03 2013
STATUS
approved