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A289078 Number of orderless same-trees of weight n.

%I A289078 #12 Aug 20 2018 20:51:15
%S A289078 1,2,2,5,2,9,2,22,6,11,2,94,2,13,12,334,2,205,2,210,14,17,2,7218,8,19,
%T A289078 68,443,2,1687,2,69109,18,23,16,167873,2,25,20,89969,2,7041,2,1548,
%U A289078 644,29,2,36094795,10,3078,24,2604,2,1484102,20,1287306,26,35,2
%N A289078 Number of orderless same-trees of weight n.
%C A289078 An orderless same-tree t is either: (case 1) a positive integer, or (case 2) a finite multiset of two or more orderless same-trees, all having the same weight. The weight of t in case 1 is the number itself, and in case 2 it is the sum of weights of the branches. For example {{{3,{1,1,1}},{2,{1,1},{1,1}}},{{{1,1,1},{1,1,1}},{{1,1},{1,1},{1,1}}}} is an orderless same-tree of weight 24 with 2 branches.
%H A289078 Alois P. Heinz, <a href="/A289078/b289078.txt">Table of n, a(n) for n = 1..2500</a>
%F A289078 a(n) = 1 + Sum_{d|n, d>1} binomial(a(n/d)+d-1, d).
%e A289078 The a(6)=9 orderless same-trees are: 6, (33), (3(111)), (222), (22(11)), (2(11)(11)), ((11)(11)(11)), ((111)(111)), (111111).
%p A289078 with(numtheory):
%p A289078 a:= proc(n) option remember; 1 + add(
%p A289078       binomial(a(n/d)+d-1, d), d=divisors(n) minus {1})
%p A289078     end:
%p A289078 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jul 05 2017
%t A289078 a[n_]:=If[n===1,1,1+Sum[Binomial[a[n/d]+d-1,d],{d,Rest[Divisors[n]]}]];
%t A289078 Array[a,100]
%o A289078 (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sumdiv(n, d, binomial(v[n/d]+d-1, d))); v} \\ _Andrew Howroyd_, Aug 20 2018
%Y A289078 Cf. A196545, A273873, A275870, A281145, A281146, A289079.
%K A289078 nonn
%O A289078 1,2
%A A289078 _Gus Wiseman_, Jun 23 2017