A029758 Number of AVL trees of height n.

1, 1, 3, 15, 315, 108675, 11878720875, 141106591466142946875, 19911070158545297149037891328865229296875, 396450714858513044552818188364610837019719636049876979456842033610756600341796875

Offset: 0

Author

Don Knuth

Keywords

• easy
• nonn

Examples

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

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..12
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A029846.

Row sums of A143897. - Alois P. Heinz, Jun 01 2009

Programs

Maple

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;

Mathematica

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 *)

PARI

{a(n) = if( n<2, n>=0, a(n-1) * (a(n-1) + 2*a(n-2)))}; /* Michael Somos, Feb 07 2004 */

Formula

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

According to Knuth (p. 715), a(n) ~ c^(2^n), where c = 1.4368728483944618758004279843355486292481149448324679771230546290458819902268... - Vaclav Kotesovec, Dec 17 2018

Extensions

More terms from N. J. A. Sloane.