0,9

A000081(m) distinct functions are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways. Some functions are representable in more than one way, the number of valid parenthesizations is A000108(m-1). The f_k are ordered, such that the number m of x's in f_k is a nondecreasing function of k. The exact ordering is defined by the algorithm below.

The list of functions f_1, f_2, ... begins:

| f_k : m : function (tree) : representation(s) : sequence |

+-----+---+------------------+--------------------------+----------+

| f_1 | 1 | x -> x | x | A019590 |

| f_2 | 2 | x -> x^x | x^x | A005727 |

| f_3 | 3 | x -> x^(x*x) | (x^x)^x | A215524 |

| f_4 | 3 | x -> x^(x^x) | x^(x^x) | A179230 |

| f_5 | 4 | x -> x^(x*x*x) | ((x^x)^x)^x | A215704 |

| f_6 | 4 | x -> x^(x^x*x) | (x^x)^(x^x), (x^(x^x))^x | A215522 |

| f_7 | 4 | x -> x^(x^(x*x)) | x^((x^x)^x) | A215705 |

| f_8 | 4 | x -> x^(x^(x^x)) | x^(x^(x^x)) | A179405 |

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Square array A(n,k) begins:

1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 1, 1, 1, 1, 1, 1, ...

0, 2, 4, 2, 6, 4, 2, 2, ...

0, 3, 12, 9, 27, 18, 15, 9, ...

0, 8, 52, 32, 156, 100, 80, 56, ...

0, 10, 240, 180, 1110, 650, 590, 360, ...

0, 54, 1188, 954, 8322, 4908, 5034, 2934, ...

0, -42, 6804, 6524, 70098, 41090, 47110, 26054, ...

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

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

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

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

end:

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

nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end

end():

A:= (n, k)-> n!*coeff(series(subs(x=x+1, f(k)), x, n+1), x, n):

seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

T[n_] := If[n == 1, {x}, Map[x^#&, g[n - 1, n - 1]]];

g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten @ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[ Range[ Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];

f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];

A[n_, k_] := n! * SeriesCoefficient[f[k] /. x -> x+1, {x, 0, n}];

Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 08 2019, after Alois P. Heinz *)

Columns k=1-17, 37 give: A019590, A005727, A215524, A179230, A215704, A215522, A215705, A179405, A215706, A215707, A215708, A215709, A215691, A215710, A215643, A215629, A179505, A211205.

Rows n=0+1, 2-10 give: A000012, A215841, A215842, A215834, A215835, A215836, A215837, A215838, A215839, A215840.

Number of distinct values taken for m x's by derivatives n=1-10: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.

Main diagonal gives A306739.

Cf. A000081, A000108, A033917, A211192, A214569, A214570, A214571, A216041, A216281, A216349, A216350, A216351, A216368, A222379, A222380, A277537, A306679, A306710, A306726.

Sequence in context: A282192 A049501 A102564 * A292712 A331571 A247504

Adjacent sequences: A215700 A215701 A215702 * A215704 A215705 A215706

sign,tabl

Alois P. Heinz, Aug 21 2012

approved