The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A215703 A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals. 58
 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 2, 12, 8, 0, 1, 1, 6, 9, 52, 10, 0, 1, 1, 4, 27, 32, 240, 54, 0, 1, 1, 2, 18, 156, 180, 1188, -42, 0, 1, 1, 2, 15, 100, 1110, 954, 6804, 944, 0, 1, 1, 8, 9, 80, 650, 8322, 6524, 38960, -5112, 0, 1, 1, 6, 48, 56, 590, 4908, 70098, 45016, 253296, 47160, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS 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  | LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened EXAMPLE 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, ... MAPLE 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); MATHEMATICA 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 *) CROSSREFS 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 KEYWORD sign,tabl AUTHOR Alois P. Heinz, Aug 21 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)