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a(n) = round(1/(1-Integral_{x=0..1} f_n(x) dx)), where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.
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%I #22 Sep 13 2019 21:41:41

%S 2,5,10,2,17,6,2,4,26,13,3,5,8,2,2,4,2,37,21,8,11,15,3,5,5,6,9,6,2,2,

%T 3,2,2,4,4,2,3,50,32,16,3,19,23,7,11,2,4,7,10,12,16,13,2,3,6,3,5,5,7,

%U 6,5,9,8,6,8,2,2,2,2,2,4,2,2,2,2,2,5,4,3,4,4

%N a(n) = round(1/(1-Integral_{x=0..1} f_n(x) dx)), where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.

%C The ordering of the functions f_n is defined in A215703: f_1, f_2, ... = x, x^x, x^(x^2), x^(x^x), x^(x^3), x^(x^x*x), x^(x^(x^2)), x^(x^(x^x)), x^(x^4), x^(x^x*x^2), ... . Values of new records are in A322008.

%H Alois P. Heinz, <a href="/A306679/b306679.txt">Table of n, a(n) for n = 1..20299</a>

%F a(n) >= 2 for n >= 1.

%p T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:

%p g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(

%p seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=

%p combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])

%p end:

%p a:= proc() local i, l; i, l:= 0, []; proc(n) while n>

%p nops(l) do i:= i+1; l:= [l[], map(f-> round(evalf(

%p 1/(1-int(f, x=0..1)))), T(i))[]] od; l[n] end

%p end():

%p seq(a(n), n=1..100);

%Y Cf. A000081, A087803, A215703, A322008.

%K nonn,look

%O 1,1

%A _Alois P. Heinz_, Mar 04 2019