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%I M0818 #57 Apr 03 2023 10:36:09
%S 2,3,6,21,231,26796,359026206,64449908476890321,
%T 2076895351339769460477611370186681,
%U 2156747150208372213435450937462082366919951682912789656986079991221
%N a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.
%C Number of nonisomorphic complete binary trees with leaves colored using two colors. - _Brendan McKay_, Feb 01 2001
%C With a(0) = 2, a(n+1) is the number of possible distinct sums between any number of elements in {1,...,a(n)}. - _Derek Orr_, Dec 13 2014
%D W. H. Cutler, Subdividing a Box into Completely Incongruent Boxes, J. Rec. Math., 12 (1979), 104-111.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A007501/b007501.txt">Table of n, a(n) for n = 0..12</a>
%H G. L. Honaker, Jr., <a href="https://t5k.org/curios/page.php?number_id=2321">41041 (another Prime Pages' Curiosity)</a>
%H J. C. Kieffer, <a href="http://dx.doi.org/10.1109/CCP.2011.36">Hierarchical Type Classes and Their Entropy Functions</a>, in 2011 First International Conference on Data Compression, Communications and Processing, pp. 246-254; Digital Object Identifier: 10.1109/CCP.2011.36.
%H J. V. Post, <a href="http://web.archive.org/web/20191228140821id_/http://www.magicdragon.com/math.html">Math Pages</a> [wayback copy]
%H Stephan Wagner, <a href="http://www.cs.sun.ac.za/~swagner/balanced.pdf">Enumeration of highly balanced trees</a>
%F a(n) = A006893(n+1) + 1.
%F a(n+1) = A000217(a(n)). - _Reinhard Zumkeller_, Aug 15 2013
%F a(n) ~ 2 * c^(2^n), where c = 1.34576817070125852633753712522207761954658547520962441996... . - _Vaclav Kotesovec_, Dec 17 2014
%e Example for depth 2 (the nonisomorphic possibilites are AAAA, AAAB, AABB, ABAB, ABBB, BBBB):
%e .........o
%e ......../.\
%e ......./...\
%e ......o.....o
%e ...../.\.../.\
%e ..../...\./...\
%e ....A...B.B...B
%t f[n_Integer] := n(n + 1)/2; NestList[f, 2, 10]
%o (PARI) a(n)=if(n<1,2,a(n-1)*(1+a(n-1))/2)
%o (Haskell)
%o a007501 n = a007501_list !! n
%o a007501_list = iterate a000217 2 -- _Reinhard Zumkeller_, Aug 15 2013
%Y Cf. A000217, A006893.
%Y Cf. A117872 (parity), A275342 (2-adic valuation).
%Y Cf. A129440.
%Y Cf. A013589 (start=4), A050542 (start=5), A050548 (start=7), A050536 (start=8), A050909 (start=9).
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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