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 A125033 Number of labeled nodes in generation n of a rooted tree where a node with label k has k child nodes with distinct labels, such that each child node is assigned the least unused label in the path that leads back to the root node with label '1'. 1
 1, 1, 2, 7, 36, 248, 2157, 22761, 283220, 4068411, 66367834, 1213504295, 24606397802 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The minimum label in generation n is n+1 and the maximum label is n*(n+1)/2 + 1. Compare to trees A029768, A007889, which seem somewhat similar; A029768 has the additional constraint that the labels must be increasing. LINKS Alec Mihailovs, A125033: Maple Program (non-recursive, uses little memory); Mapleprimes, Nov 19 2006. EXAMPLE Labels for the initial nodes of the tree for generations 0..4: gen 0: [1]; gen 1: (1)->[2]; gen 2: (2)->[3,4]; gen 3: (3)->[4,5,6], (4)->[3,5,6,7]; gen 4: (4)->[5,6,7,8], (5)->[4,6,7,8,9], (6)->[4,5,7,8,9,10], (3)->[5,6,7], (5)->[3,6,7,8,9], (6)->[3,5,7,8,9,10], (7)->[3,5,6,8,9,10,11]; MAPLE gen := proc(parents, maxgen, ocounts, lvl) local thislbl, lbl, childlbl, counts, npar; counts := ocounts; counts[lvl] := counts[lvl]+1; if nops(parents) < maxgen then thislbl := op(-1, parents); childlbl := 1; for lbl from 1 to thislbl do while ( childlbl in parents ) or ( childlbl = thislbl ) do childlbl := childlbl+1; od; npar := [op(parents), childlbl]; if nops(counts) < lvl+1 then counts := [op(counts), 0]; fi; counts := gen(npar, maxgen, counts, lvl+1); childlbl := childlbl+1; od; fi; if lvl <= maxgen -4 then print(counts); fi; RETURN(counts); end: maxgen := 8; parents := [1, 2]; n := [1, 0]; gen(parents, maxgen, n, 2); print(%) ; # R. J. Mathar, Nov 17 2006) The following Maple10 code is from Alec Mihailovs: # (start) f:=proc(n::integer[4], A::Array(datatype=integer[4]), B::Array(datatype=integer[4]))::integer[4]; local c::integer[4], i::integer[4], len::integer[4], m::integer[4]; c, len, m:=0, 3, 3; while len>1 do if len=n then c:=c+1; m:=A[len]; B[m]:=0; len:=len-1; B[A[len]]:=B[A[len]]+1 elif B[A[len]]<=A[len] then for i from m+1 do if B[i]=0 then break fi od; len:=len+1; A[len]:=i; B[i]:=1; m:=2 else m:=A[len]; B[m]:=0; len:=len-1; B[A[len]]:=B[A[len]]+1 fi od; c end: cf:=Compiler:-Compile(f): F:=proc(n::posint) local A, B; if n<3 then 1 elif n=3 then 2 else A:=Array([\$1..3, 0\$(n-3)], datatype=integer[4]); B:=Array([1\$3, 0\$((n-2)*(n+1)/2)], datatype=integer[4]); cf(n, A, B) fi end: seq(F(n), n=1..12); # (end) CROSSREFS Sequence in context: A180271 A167199 A007889 * A034430 A143805 A249637 Adjacent sequences:  A125030 A125031 A125032 * A125034 A125035 A125036 KEYWORD nonn AUTHOR Paul D. Hanna and R. J. Mathar, Nov 17 2006 EXTENSIONS Terms a(6)-a(10) from R. J. Mathar, Nov 17 2006 a(11) and a(12) from Alec Mihailovs (alec(AT)mihailovs.com), Nov 19 2006 STATUS approved

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Last modified February 20 13:14 EST 2020. Contains 332077 sequences. (Running on oeis4.)