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A318049 Number of first/rest balanced rooted plane trees with n nodes. 4
1, 0, 1, 0, 1, 1, 1, 3, 2, 6, 8, 11, 26, 28, 67, 96, 162, 316, 448, 922 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..20.

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}]

CROSSREFS

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

Sequence in context: A131006 A122362 A072635 * A210754 A210738 A210601

Adjacent sequences:  A318046 A318047 A318048 * A318050 A318051 A318052

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 13 2018

STATUS

approved

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Last modified November 18 16:23 EST 2018. Contains 317323 sequences. (Running on oeis4.)