A110387 a(1) = 1, a(2) = a(1) + 1, a(3) = a(2)^2 + a(1) + 1; a(n+1) = a(n)^n + a(n-1)^(n-1) + ... + a(2)^2 + a(1) + 1.

1, 2, 6, 222, 2428912878, 84539502447168140812774402430429967456348471246

Offset

1,2

Comments

The next term is too large to include.

a(n) = the number of ordered trees with root degree at most n-1 and having strictly thinning limbs. An ordered tree with strictly thinning limbs is such that if a vertex has k children, each of its children has fewer than k children. For example, we have a(1) = 1 (the 1-vertex tree) and a(2)=1 (the 1-vertex tree and the 1-edge tree). From here one obtains easily (i) the recurrence relation for a(n+1) given in NAME (the terms in the right-hand side count successively the trees with root degrees n-1, n-2, ..., 1, 0, respectively) and also (ii) the recurrence relation for a(n) given in FORMULA (the first term in the right-hand side counts the trees with root degree at most n-2 and the second term counts the trees with root degree n-1). Moreover, it follows that the terms of this sequence are the partial sums of the sequence obtained from A248099 by extending it with A248099(0) = 1. - Emeric Deutsch, Aug 11 2015

Links

Table of n, a(n) for n=1..6.

Formula

a(n) = a(n-1) + (a(n-1))^(n-1); a(1) = 1. The 2nd Maple program is based on this. - Emeric Deutsch, Jan 08 2015

Example

a(3) = 2^2 + 1 + 1 = 6.
a(4) = 6^3 + 2^2 + 1^1 + 1 = 222.

Maple

a[1]:=1: for n from 1 to 6 do a[n+1]:=1+sum(a[j]^j, j=1..n) od: seq(a[n], n=1..7); # Emeric Deutsch, Jul 31 2005
a[1]:=1: for n from 2 to 7 do a[n]:=a[n-1]+a[n-1]^(n-1) end do: seq(a[n], n = 1 .. 7); # Emeric Deutsch, Jan 08 2015

Mathematica

A110387 = {}; s = 1; Do[s += s^n; AppendTo[A110387, s], {n, 3!}]; A110387 (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)

Prog

(Magma) [n eq 1 select 1 else Self(n-1)+Self(n-1)^(n-1): n in [1..7]]; // Vincenzo Librandi, Aug 12 2015

Crossrefs

Cf. A248099.

Sequence in context: A176782 A357191 A013083 * A158682 A201626 A332692

Adjacent sequences: A110384 A110385 A110386 * A110388 A110389 A110390

Keyword

easy,nonn

Author

Amarnath Murthy, Jul 26 2005

Extensions

Corrected and extended by Emeric Deutsch and Erich Friedman, Jul 31 2005

Status

approved