A298539 -id:A298539 - cp's OEIS frontend

A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

1, 1, 2, 3, 6, 10, 25, 49, 127, 291, 766, 1843, 5003, 12487, 34151, 87983, 242088, 634848, 1763749, 4688677, 13085621, 35241441, 98752586, 268282856, 755353825, 2067175933, 5837592853, 16087674276, 45550942142, 126186554309, 358344530763, 997171512999

Offset: 1

Gus Wiseman, Jan 20 2018

nonn

			The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), ((o)(o)), (oooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A003238, A004111, A032305, A289078, A289079, A290689, A291443, A297791, A298422, A298533, A298535, A298538, A298539.

r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[If[n===1,1,Sum[Binomial[r[(n-1)/d]+d-1,d],{d,Divisors[n-1]}]],{n,40}]

a(n + 1) = Sum_{d|n} binomial(A000081(n/d) + d - 1, d).

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

A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

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