OFFSET
1,8
COMMENTS
A rooted plane tree is first/rest balanced if either (1) it is a single node, or (2a) the number of leaves in the first branch is equal to the number of branches minus one, and (2b) every branch is also first/rest balanced.
Also the number of composable free pure multifunctions (CPMs) with one atom and n positions. A CPM is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h and each of the g_i for i = 1, ..., k > 0 are CPMs, and the number of leaves in h is equal to k. The number of positions in a CPM is the number of brackets [...] plus the number of o's.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..100
FORMULA
G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y + Sum_{k>=1} y^k * ([y^k] A(x,y)) * A(x,y)^k). - Andrew Howroyd, Jan 22 2021
EXAMPLE
The a(12) = 11 first/rest balanced rooted plane trees:
(o(o(o((oo)oo))))
(o(o((oo)(oo)o)))
(o(o((oo)o(oo))))
(o((oo)(o(oo))o))
(o((oo)o(o(oo))))
(o((oo)(oo)(oo)))
((oo)(o(o(oo)))o)
((oo)o(o(o(oo))))
((o(o(oo)))oooo)
((oo)(o(oo))(oo))
((oo)(oo)(o(oo)))
The a(12) = 11 composable free pure multifunctions:
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]]]]
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]]]
o[o[o[o]]][o,o,o,o]
MATHEMATICA
balplane[n_]:=balplane[n]=If[n===1, {{}}, Join@@Function[c, Select[Tuples[balplane/@c], Length[Cases[#[[1]], {}, {0, Infinity}]]==Length[#]-1&]]/@Join@@Permutations/@IntegerPartitions[n-1]];
Table[Length[balplane[n]], {n, 10}]
PROG
(PARI) seq(n)={my(p=x*y+O(x^2)); for(n=1, n\2, p = x*y + x*sum(k=1, n, y^k * polcoef(p, k, y) * (O(x^(2*n-k+1)) + p)^k )); Vec(subst(p + O(x*x^n), y, 1)) } \\ Andrew Howroyd, Jan 22 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 13 2018
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Jan 22 2021
STATUS
approved