login
This site is supported by donations to The OEIS Foundation.

 

Logo


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. 54
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

with(combinat):

T:= proc(n) T(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(T(i)[w[t]-t+1], t=1..j)*v,

      w=choose([$1..nops(T(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)])

    end:

f:= proc() local b, i, l; i, l:= 0, []; b:= proc(n) option remember;

      while nops(l)<n 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);

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.

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

Sequence in context: A282192 A049501 A102564 * A292712 A247504 A235955

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 22 11:07 EDT 2018. Contains 316438 sequences. (Running on oeis4.)