A298479 -id:A298479 - cp's OEIS frontend

A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.

1, 1, 1, 3, 3, 5, 13, 15, 23, 34, 95, 106, 176, 241, 374, 942, 1129, 1760, 2515, 3711, 5136, 12857, 14911, 23814, 33002, 49141, 65798, 97056, 209707, 255042, 389725, 545290, 790344, 1071010, 1525919, 2043953, 4272124, 5110583, 7772247, 10611491, 15447864, 20496809

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

a(n) is the number of labeled trees with sum of the labels equal to n-1 and the outdegree of every node less than or equal to the value of its label. - Andrew Howroyd, Feb 02 2021

			The a(7) = 13 trees: ((o(ooo))), ((oo(oo))), ((ooooo)), (o((ooo))), (o(oo(o))), (o(oooo)), ((o)(ooo)), (oo((oo))), (oo(o(o))), (o(o)(oo)), (ooo(oo)), (oooo(o)), (oooooo).

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

Cf. A000081, A001190, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298304, A298422, A298479.

krut[n_]:=krut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[krut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@Length/@Cases[#,{},{0,Infinity}]&]];
Table[krut[n]//Length,{n,15}]

relabel(b)={my(w=hammingweight(b)); b = bitand((1<Andrew Howroyd, Feb 02 2021

a(27)-a(34) from Robert G. Wilson v, Jan 19 2018

Terms a(35) and beyond from Andrew Howroyd, Feb 02 2021

A298540 Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, Jan 21 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
6  (o(o))
7  ((oo))
10 (o((o)))
11 ((((o))))
13 ((o(o)))
14 (o(oo))
15 ((o)((o)))
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
22 (o(((o))))
23 (((o)(o)))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298479, A298534, A298536, A298538, A298539.

nn=500;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[MGweight/@primeMS[n]]];
Select[Range[nn],UnsameQ@@MGweight/@primeMS[#]&]

cp's OEIS Frontend

A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.

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