A318049 Number of first/rest balanced rooted plane trees with n nodes.

1, 0, 1, 0, 1, 1, 1, 3, 2, 6, 8, 11, 26, 28, 67, 96, 162, 316, 448, 922, 1435, 2572, 4660, 7563, 14397, 23896, 43337, 77097, 133071, 244787, 423093, 767732, 1367412, 2426612, 4408497, 7802348, 14152342, 25365035, 45602031, 82631362, 148246136, 269103870, 485379304

Offset: 1

Author

Gus Wiseman, Aug 13 2018

Keywords

• nonn

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.

Examples

			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]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000081, A000108, A001003, A001006, A007853, A126120, A317713, A318046, A318048.

Programs

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}]

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

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

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 22 2021