A001697 a(n+1) = a(n)(a(0) + ... + a(n)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• 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

Crossrefs

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

Cf. A002658, A001699.

Cf. A064847.

Programs

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

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 *)

PARI

a(n)=if(n<2,n >= 0,a(n-1)^2*(1+1/a(n-2)))

Formula

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

Extensions

Additional comments from Michael Somos, May 19 2000