cp's OEIS Frontend

A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

%I A199296 #20 Sep 01 2023 10:14:21
%S A199296 1,1,2,4,9,20,45,92,182,342,601,982,1499,2169,2970,3994,5297,6834,
%T A199296 8635,10714,13121,16104,19674,23868,28453,33637,39630,46730
%N A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
%e A199296 a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.
%p A199296 f:= proc(n) option remember;
%p A199296       `if`(n=1, {[0, 0, 0, 0]},
%p A199296             {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
%p A199296              8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
%p A199296              10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
%p A199296              +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
%p A199296               h=f(n-j)), g=f(j)), j=1..n-1)})
%p A199296     end:
%p A199296 a:= n-> nops(map(x-> x[4], f(n))):
%p A199296 seq(a(n), n=1..20);
%t A199296 f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
%t A199296 a[n_] := Length@Union@(#[[4]]& /@ f[n]);
%t A199296 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* _Jean-François Alcover_, Sep 01 2023, after _Alois P. Heinz_ *)
%Y A199296 Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215835. Column k=5 of A216368.
%K A199296 nonn
%O A199296 1,3
%A A199296 _Alois P. Heinz_, Nov 04 2011