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A306884 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) decreasing order.

%I A306884 #22 Feb 16 2025 08:33:55
%S A306884 1,1,3,6,14,28,93,270,86170,7625640881546
%N A306884 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) decreasing order.
%C A306884 a(10) = 200352993...611306920 has 19729 decimal digits.
%H A306884 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A306884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponentiation">Exponentiation</a>
%H A306884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Identity_element">Identity element</a>
%H A306884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Operator_associativity">Operator associativity</a>
%H A306884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A306884 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 A306884 a(6) = 1^1^1^1^1^1 + 2^1^1^1^1 + 2^2^1^1 + 2^2^2 + 3^1^1^1 + 3^2^1 + 3^3 + 4^1^1 + 4^2 + 5^1 + 6 = 1 + 2 + 4 + 16 + 3 + 9 + 27 + 4 + 16 + 5 + 6 = 93.
%p A306884 f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
%p A306884 a:= n-> add(f(sort(l, `>`)), l=combinat[partition](n)):
%p A306884 seq(a(n), n=0..9);
%Y A306884 Cf. A006906, A066186, A306895, A306901, A306902, A306903, A306918.
%K A306884 nonn
%O A306884 0,3
%A A306884 _Alois P. Heinz_, Mar 15 2019