A306918 - OEIS

A306918

Sum over all partitions of n into distinct parts of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in decreasing order.

1, 1, 2, 5, 7, 18, 36, 118, 265, 263212, 2217881, 152599933940, 542101086242752217003726400434973829461152534, 63340828764059520458379290673240751904836319648345

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(14) = 620606987...270037949 has 183231 decimal digits.

LINKS

Table of n, a(n) for n=0..13.

Eric Weisstein's World of Mathematics, Power Tower

Wikipedia, Exponentiation

Wikipedia, Identity element

Wikipedia, Operator associativity

Wikipedia, Partition (number theory)

EXAMPLE

a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.

a(6) = 3^2^1 + 4^2 + 5^1 + 6 = 9 + 16 + 5 + 6 = 36.

MAPLE

d:= proc(l) local i; for i to nops(l)-1 do

if l[i]=l[i+1] then return fi od; l

end:

f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):

a:= n-> add(f(l), l=map(l->d(sort(l, `>`)), combinat[partition](n))):

seq(a(n), n=0..13);

MATHEMATICA

d[l_] := Module[{i}, For[i = 1, i <= Length[l] - 1, i++, If[l[[i]] == l[[i + 1]], Return[]]]; l];

f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];

a[n_] := Sum[f[l], {l, ReverseSort /@ Select[IntegerPartitions[n], Length@# == Length@ Union@# &]}];