A001697 a(n+1) = a(n)(a(0) + ... + a(n)). (Formerly M1902 N0751)

1, 1, 2, 8, 96, 10368, 108615168, 11798392572168192, 139202068568601556987554268864512, 19377215893777651167043206536157390321290709180447278572301746176

Offset

0,3

Comments

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

The next term (a(10)) has 129 digits. - Harvey P. Dale, Jan 24 2016

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

References

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).

Links

John Cerkan, Table of n, a(n) for n = 0..12

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, alternative link.

Daniel Duverney, Takeshi Kurosawa, Iekata Shiokawa, Transformation formulas of finite sums into continued fractions, arXiv:1912.12565 [math.NT], 2019.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Index to divisibility sequences

Formula

a(n) ~ c^(2^n), where c = 1.3352454783981919948826893254756974184778316104856161827213437094446034867599... . - Vaclav Kotesovec, May 21 2015

Mathematica

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 *)

Prog

(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
-- Reinhard Zumkeller, Apr 29 2013
(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

Crossrefs

a(n) = A039941(2*n+1); first differences of A001696 give this sequence.

Cf. A002658, A001699.

Cf. A064847.

Sequence in context: A156926 A361388 A326866 * A006069 A270485 A223042

Adjacent sequences: A001694 A001695 A001696 * A001698 A001699 A001700

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Somos, May 19 2000

Status

approved