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

Offset: 0

Author

Alois P. Heinz, Mar 16 2019

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Exponentiation
• Wikipedia, Identity element
• Wikipedia, Operator associativity
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A022629, A066189, A306884, A306919.

Programs

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@# &]}];
a /@ Range[0, 13] (* Jean-François Alcover, May 04 2020, after Maple *)