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A001697
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a(n+1) = a(n)(a(0) + ... + a(n)).
(Formerly M1902 N0751)
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5
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OFFSET
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0,3
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COMMENTS
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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
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
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) ~ c^(2^n), where c = 1.3352454783981919948826893254756974184778316104856161827213437094446034867599... . - Vaclav Kotesovec, May 21 2015
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MATHEMATICA
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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 *)
nxt[{t_, a_}]:={t+t*a, t*a}; Transpose[NestList[nxt, {1, 1}, 10]][[2]] (* Harvey P. Dale, Jan 24 2016 *)
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PROG
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(PARI) a(n)=if(n<2, n >= 0, a(n-1)^2*(1+1/a(n-2)))
(Haskell)
a001697 n = a001697_list !! n
a001697_list = 1 : 1 : f [1, 1] where
f xs@(x:_) = y : f (y : xs) where y = x * sum xs
(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
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CROSSREFS
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a(n) = A039941(2*n+1); first differences of A001696 give this sequence.
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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