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A306895 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) increasing order.

%I A306895 #19 Feb 16 2025 08:33:55
%S A306895 1,1,3,5,11,18,72,387,134349386,
%T A306895 115792089237316195423570985008687907853269984665640566457309223244801371506483
%N A306895 Sum over all partitions of n of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in (weakly) increasing order.
%C A306895 a(10) has 40403562 decimal digits.
%H A306895 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A306895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponentiation">Exponentiation</a>
%H A306895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Identity_element">Identity element</a>
%H A306895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Operator_associativity">Operator associativity</a>
%H A306895 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A306895 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 A306895 a(6) = 1^1^1^1^1^1 + 1^1^1^1^2 + 1^1^2^2 + 2^2^2 + 1^1^1^3 + 1^2^3 + 3^3 + 1^1^4 + 2^4 + 1^5 + 6 = 1 + 1 + 1 + 16 + 1 + 1 + 27 + 1 + 16 + 1 + 6 = 72.
%p A306895 f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
%p A306895 a:= n-> add(f(sort(l, `<`)), l=combinat[partition](n)):
%p A306895 seq(a(n), n=0..9);
%Y A306895 Cf. A006906, A066186, A306884, A306901, A306902, A306903, A306919.
%K A306895 nonn
%O A306895 0,3
%A A306895 _Alois P. Heinz_, Mar 15 2019