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%I #27 Nov 14 2023 12:13:33
%S 1,2,21,392,11980,471966,24655820,1548264752,118039822488,
%T 10482116888640,1076582148812808,125439212178037728,
%U 16473767684928836256,2410412979008498588208,390793360308270931979400,69716064087131957637475968
%N Sum of n-th derivatives at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways.
%e For n=4 the A000081(4) = 4 functions and their 4th derivatives at x=1 are x^(x^3)->156, x^(x^x*x)->100, x^(x^(x^2))->80, x^(x^(x^x))->56 => a(n) = 156+100+80+56 = 392.
%p F:= proc(n) F(n):= `if`(n=1, [x], map(h->x^h, g(n-1, n-1))) end:
%p g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(seq(
%p mul(F(i)[w[t]-t+1], t=1..j)*v, w=combinat[choose](
%p [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
%p end:
%p a:= n-> add(n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), f=F(n)):
%p seq(a(n), n=1..12);
%t F[n_] := If[n == 1, {x}, x^#& /@ g[n - 1, n - 1]];
%t g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten@Table[Table[Table[ Product[F[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {w, Subsets[Range[Length[ F[i]] + j - 1], {j}]}], {v, g[n - i*j, i - 1]}], {j, 0, n/i}]];
%t a[n_] := a[n] = Sum[n!*SeriesCoefficient[f /. x -> x + 1, {x, 0, n}], {f, F[n]}];
%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 12}] (* _Jean-François Alcover_, Nov 14 2023, after _Alois P. Heinz_ *)
%Y Row sums of A216349, A216350.
%Y Cf. A000081, A215703.
%K nonn
%O 1,2
%A _Alois P. Heinz_, Sep 04 2012
%E a(16) from _Alois P. Heinz_, May 09 2016
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