A306919 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 increasing order.

1, 1, 2, 4, 5, 14, 24, 122, 318, 2417851639229258349414245, 14134776518227074636666380005943348126619871175004951664972849610340964762

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..11

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) = 1^2^3 + 2^4 + 1^5 + 6 = 1 + 16 + 1 + 6 = 24.

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..11);

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, Sort /@ Select[IntegerPartitions[n], Length@# == Length @ Union@#&]}];
a /@ Range[0, 11] (* Jean-François Alcover, May 03 2020, after Maple *)

Crossrefs

Cf. A022629, A066189, A306895, A306918.

Sequence in context: A295402 A121410 A002959 * A373950 A295031 A178436

Adjacent sequences: A306916 A306917 A306918 * A306920 A306921 A306922

Keyword

nonn

Author

Alois P. Heinz, Mar 16 2019

Status

approved