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A061773
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Triangle in which n-th row lists Matula-Goebel numbers for all rooted trees with n nodes.
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35
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 15, 18, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 53, 59, 67, 25, 27, 30, 33, 35, 36, 39, 40, 42, 44, 46, 47, 48, 49, 51, 52, 56, 57, 58, 61, 62, 64, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 101, 106
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refs;
listen;
history;
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OFFSET
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1,2
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COMMENTS
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Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.
First entry in row n is A005517(n).
The Maple program yields row n after defining F = A005517(n) and L = A005518(n).
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LINKS
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EXAMPLE
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The labels for the rooted trees with at most 4 nodes are as follows (x is the root):
o
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o o o o o
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o o o o o o o o o o o
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x x x x x x x x
1 2 4 3 8 6 7 5 (label)
Triangle begins:
1;
2;
3,4;
5,6,7,8;
9,10,11,12,13,14,16,17,19;
15,18,20,21,22,23,24,26,28,29,31,32,34,37,38,41,43,53,59,67;
25,27,30,33,35,36,39,40,42,44,46,47,48,49,51,52,56,57,58,61,62,64,68,\
71,73,74,76,79,82,83,86,89,101,106,107,109,118,127,131,134,139,157,163,\
179,191,241,277,331;
...
Triangle of rooted trees represented as finitary multisets begins:
(),
(()),
((())), (()()),
(((()))), (()(())), ((()())), (()()()),
((())(())), (()((()))), ((((())))), (()()(())), ((()(()))), (()(()())), (()()()()), (((()()))), ((()()())). - Gus Wiseman, Dec 21 2016
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MAPLE
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n := 8: F := 45: L := 2221: with(numtheory): N := proc (m) local r, s: r := proc (m) options operator, arrow: op(1, factorset(m)) end proc: s := proc (m) options operator, arrow: m/r(m) end proc: if m = 1 then 1 elif bigomega(m) = 1 then 1+N(pi(m)) else N(r(m))+N(s(m))-1 end if end proc: A := {}: for k from F to L do if N(k) = n then A := `union`(A, {k}) else end if end do: A;
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MATHEMATICA
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F[n_] := F[n] = Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^(n/3-1), Mod[n, 3] == 1, 5^(n/3-1/3), True, 9*5^(n/3-5/3)]; L[n_] := L[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, _, Prime[L[n-1]]]; r[m_] := FactorInteger[m][[1, 1]]; s[m_] := m/r[m]; NN[m_] := NN[m] = Which[m == 1, 1, PrimeOmega[m] == 1, 1+NN[PrimePi[m]], True, NN[r[m]]+NN[s[m]]-1]; row[n_] := Module[{A, k}, A = {}; For[k = F[n], k <= L[n], k++, If[NN[k] == n, A = Union[A, {k}]]]; A]; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Maple *)
nn=8; MGweight[n_]:=If[n===1, 1, 1+Total[Cases[FactorInteger[n], {p_, k_}:>k*MGweight[PrimePi[p]]]]];
Take[GatherBy[Range[Switch[nn, 1, 1, 2, 2, 3, 4, _, Nest[Prime, 8, nn-4]]], MGweight], nn] (* Gus Wiseman, Dec 21 2016 *)
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PROG
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(PARI) See links.
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CROSSREFS
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KEYWORD
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nonn,tabf,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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