%I M1902 N0751 #60 Apr 19 2024 17:39:15
%S 1,1,2,8,96,10368,108615168,11798392572168192,
%T 139202068568601556987554268864512,
%U 19377215893777651167043206536157390321290709180447278572301746176
%N a(n+1) = a(n)(a(0) + ... + a(n)).
%C Number of binary trees of height n where for each node the left subtree is at least as high as the right subtree. - _Franklin T. Adams-Watters_, Feb 08 2007
%C The next term (a(10)) has 129 digits. - _Harvey P. Dale_, Jan 24 2016
%C Number of plane trees where the root has exactly n children and the i-th child of any node has at most i-1 children. - _David Eppstein_, Dec 18 2021
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H John Cerkan, <a href="/A001697/b001697.txt">Table of n, a(n) for n = 0..12</a>
%H A. V. Aho and N. J. A. Sloane, <a href="https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.
%H Daniel Duverney, Takeshi Kurosawa, Iekata Shiokawa, <a href="https://arxiv.org/abs/1912.12565">Transformation formulas of finite sums into continued fractions</a>, arXiv:1912.12565 [math.NT], 2019.
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) ~ c^(2^n), where c = 1.3352454783981919948826893254756974184778316104856161827213437094446034867599... . - _Vaclav Kotesovec_, May 21 2015
%t a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1]^2*(1 + 1/a[n - 2]); Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Jul 02 2013 *)
%t nxt[{t_,a_}]:={t+t*a,t*a}; Transpose[NestList[nxt,{1,1},10]][[2]] (* _Harvey P. Dale_, Jan 24 2016 *)
%o (PARI) a(n)=if(n<2,n >= 0,a(n-1)^2*(1+1/a(n-2)))
%o (Haskell)
%o a001697 n = a001697_list !! n
%o a001697_list = 1 : 1 : f [1,1] where
%o f xs@(x:_) = y : f (y : xs) where y = x * sum xs
%o -- _Reinhard Zumkeller_, Apr 29 2013
%o (Magma) [n le 2 select 1 else Self(n-1)^2*(1+1/Self(n-2)): n in [1..12]]; // _Vincenzo Librandi_, Nov 25 2015
%Y a(n) = A039941(2*n+1); first differences of A001696 give this sequence.
%Y Cf. A002658, A001699.
%Y Cf. A064847.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, May 19 2000