%I #30 Dec 17 2018 10:35:20

%S 1,1,3,15,315,108675,11878720875,141106591466142946875,

%T 19911070158545297149037891328865229296875,

%U 396450714858513044552818188364610837019719636049876979456842033610756600341796875

%N Number of AVL trees of height n.

%D D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

%H Alois P. Heinz, <a href="/A029758/b029758.txt">Table of n, a(n) for n = 0..12</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F a(n+1) = a(n)^2 + 2*a(n)*a(n-1).

%F According to Knuth (p. 715), a(n) ~ c^(2^n), where c = 1.4368728483944618758004279843355486292481149448324679771230546290458819902268... - _Vaclav Kotesovec_, Dec 17 2018

%e G.f. = 1 + x + 3*x^2 + 15*x^3 + 315*x^4 + 108675*x^5 + 11878720875*x^6 + ...

%p A029758 := proc(n) option remember; if n <= 1 then RETURN(1); else A029758(n-1)^2+2*A029758(n-1)*A029758(n-2); fi; end;

%t a[0] = a[1] = 1; a[n_] := a[n] = a[n-1]^2 + 2*a[n-1]*a[n-2]; Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Feb 13 2015 *)

%o (PARI) {a(n) = if( n<2, n>=0, a(n-1) * (a(n-1) + 2*a(n-2)))}; /* _Michael Somos_, Feb 07 2004 */

%Y Cf. A029846.

%Y Row sums of A143897. - _Alois P. Heinz_, Jun 01 2009

%K easy,nonn

%O 0,3

%A _Don Knuth_

%E More terms from _N. J. A. Sloane_.