A306919 -id:A306919 - cp's OEIS frontend

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.

1, 1, 3, 5, 11, 18, 72, 387, 134349386, 115792089237316195423570985008687907853269984665640566457309223244801371506483

Offset: 0

Alois P. Heinz, Mar 15 2019

nonn

a(10) has 40403562 decimal digits.

			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 + 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.

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

Cf. A006906, A066186, A306884, A306901, A306902, A306903, A306919.

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);

A306918 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.

Original entry on oeis.org

1, 1, 2, 5, 7, 18, 36, 118, 265, 263212, 2217881, 152599933940, 542101086242752217003726400434973829461152534, 63340828764059520458379290673240751904836319648345

Offset: 0

Alois P. Heinz, Mar 16 2019

nonn

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

			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) = 3^2^1 + 4^2 + 5^1 + 6 = 9 + 16 + 5 + 6 = 36.

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

Cf. A022629, A066189, A306884, A306919.

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..13);

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, ReverseSort /@ Select[IntegerPartitions[n], Length@# == Length@ Union@# &]}];
a /@ Range[0, 13] (* Jean-François Alcover, May 04 2020, after Maple *)

cp's OEIS Frontend

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.

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