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.

1, 1, 3, 6, 14, 28, 93, 270, 86170, 7625640881546

Offset: 0

Author

Alois P. Heinz, Mar 15 2019

Keywords

• nonn

Comments

a(10) = 200352993...611306920 has 19729 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) = 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.
		

Links

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

Crossrefs

Cf. A006906, A066186, A306895, A306901, A306902, A306903, A306918.

Programs

Maple

f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(sort(l, `>`)), l=combinat[partition](n)):
seq(a(n), n=0..9);