OFFSET
1,4
COMMENTS
In other words, rooted trees with all leaves at the same level and no node having exactly one child; the order of children is not significant.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
Let s_0(n) = 1 if n = 1, 0 otherwise; s_{k+1}(n) = EULER(s_k)(n) - s_k(n), where EULER is the Euler transform. Then a_n = sum_k s_k(n). (s_k(n) is the number of such trees of height k.) Note that s_k(n) = 0 for n < 2^k.
EXAMPLE
From Gus Wiseman, Oct 07 2018: (Start)
The a(10) = 16 series-reduced balanced rooted trees:
(oooooooooo)
((ooooo)(ooooo))
((oooo)(oooooo))
((ooo)(ooooooo))
((oo)(oooooooo))
((ooo)(ooo)(oooo))
((oo)(oooo)(oooo))
((oo)(ooo)(ooooo))
((oo)(oo)(oooooo))
((oo)(oo)(ooo)(ooo))
((oo)(oo)(oo)(oooo))
((oo)(oo)(oo)(oo)(oo))
(((oo)(ooo))((oo)(ooo)))
(((oo)(oo))((ooo)(ooo)))
(((oo)(oo))((oo)(oooo)))
(((oo)(oo))((oo)(oo)(oo)))
(End)
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(u=vector(n), v=vector(n)); u[1]=1; while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Aug 18 2006
STATUS
approved