closed form for A007501

Brendan McKay bdm at cs.anu.edu.au
Wed Jan 31 13:29:10 CET 2001


%I A007501 M0818
%S A007501 2,3,6,21,231,26796,359026206,64449908476890321,
%T A007501 2076895351339769460477611370186681,
%U A007501 2156747150208372213435450937462082366919951682912789656986079991221
%N A007501 a(n+1) = a(n)*(a(n)+1)/2.
%D A007501 W. H. Cutler, Subdividing a Box into Completely Incongruent Boxes, J.  Rec. Math., 12 (1979), 104-111.
%o A007501 (PARI) a(n)=if(n<1,2,a(n-1)*(1+a(n-1))/2)
%t A007501 f[n_Integer] := n(n + 1)/2; NestList[f, 2, 10]
%K A007501 nonn,easy,huge                 
%O A007501 0,1
%A A007501 njas, Robert G. Wilson v (rgwv at kspaint.com)

I haven't looked at that reference, so I don't know if the following
alternative description is equivalent:

   The number of nonisomorphic complete binary trees
   with leaves coloured using two colours.

Example for depth 2:

         o
        / \
       /   \
      o     o
     / \   / \
    /   \ /   \     The nonisomorpic possibilites are
    A   B B   B     AAAA, AAAB, AABB, ABAB, ABBB, BBBB.

My real question: is there a closed form for this sequence?

Brendan.





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