A216281 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.

1, 2, 21, 392, 11980, 471966, 24655820, 1548264752, 118039822488, 10482116888640, 1076582148812808, 125439212178037728, 16473767684928836256, 2410412979008498588208, 390793360308270931979400, 69716064087131957637475968

Offset

1,2

Links

Table of n, a(n) for n=1..16.

Example

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.

Maple

F:= proc(n) F(n):= `if`(n=1, [x], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(seq(
       mul(F(i)[w[t]-t+1], t=1..j)*v, w=combinat[choose](
       [$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])
    end:
a:= n-> add(n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), f=F(n)):
seq(a(n), n=1..12);

Mathematica

F[n_] := If[n == 1, {x}, x^#& /@ g[n - 1, n - 1]];
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}]];
a[n_] := a[n] = Sum[n!*SeriesCoefficient[f /. x -> x + 1, {x, 0, n}], {f, F[n]}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Nov 14 2023, after Alois P. Heinz *)

Crossrefs

Row sums of A216349, A216350.

Cf. A000081, A215703.

Sequence in context: A099710 A098344 A228384 * A263625 A245686 A091315

Adjacent sequences: A216278 A216279 A216280 * A216282 A216283 A216284

Keyword

nonn

Author

Alois P. Heinz, Sep 04 2012

Extensions

a(16) from Alois P. Heinz, May 09 2016

Status

approved