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%I #11 Jan 22 2021 15:43:57
%S 1,0,1,0,1,1,1,3,2,6,8,11,26,28,67,96,162,316,448,922,1435,2572,4660,
%T 7563,14397,23896,43337,77097,133071,244787,423093,767732,1367412,
%U 2426612,4408497,7802348,14152342,25365035,45602031,82631362,148246136,269103870,485379304
%N Number of first/rest balanced rooted plane trees with n nodes.
%C 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.
%C 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.
%H Andrew Howroyd, <a href="/A318049/b318049.txt">Table of n, a(n) for n = 1..100</a>
%F 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
%e The a(12) = 11 first/rest balanced rooted plane trees:
%e (o(o(o((oo)oo))))
%e (o(o((oo)(oo)o)))
%e (o(o((oo)o(oo))))
%e (o((oo)(o(oo))o))
%e (o((oo)o(o(oo))))
%e (o((oo)(oo)(oo)))
%e ((oo)(o(o(oo)))o)
%e ((oo)o(o(o(oo))))
%e ((o(o(oo)))oooo)
%e ((oo)(o(oo))(oo))
%e ((oo)(oo)(o(oo)))
%e The a(12) = 11 composable free pure multifunctions:
%e o[o[o[o[o][o,o]]]]
%e o[o[o[o][o[o],o]]]
%e o[o[o[o][o,o[o]]]]
%e o[o[o][o[o[o]],o]]
%e o[o[o][o,o[o[o]]]]
%e o[o[o][o[o],o[o]]]
%e o[o][o[o[o[o]]],o]
%e o[o][o,o[o[o[o]]]]
%e o[o][o[o[o]],o[o]]
%e o[o][o[o],o[o[o]]]
%e o[o[o[o]]][o,o,o,o]
%t 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]];
%t Table[Length[balplane[n]],{n,10}]
%o (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
%Y Cf. A000081, A000108, A001003, A001006, A007853, A126120, A317713, A318046, A318048.
%K nonn
%O 1,8
%A _Gus Wiseman_, Aug 13 2018
%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 22 2021
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