A306918 - OEIS

%I #18 May 04 2020 16:29:57

%S 1,1,2,5,7,18,36,118,265,263212,2217881,152599933940,

%T 542101086242752217003726400434973829461152534,

%U 63340828764059520458379290673240751904836319648345

%N 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.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponentiation">Exponentiation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Identity_element">Identity element</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Operator_associativity">Operator associativity</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

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

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

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

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

%p     end:

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

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

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

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

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

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

%t a /@ Range[0, 13] (* _Jean-François Alcover_, May 04 2020, after Maple *)

%Y Cf. A022629, A066189, A306884, A306919.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 16 2019