cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2012

Keywords

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 |

Examples

			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, ...

Links

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.

Programs

  • 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 *)

A216368 Number T(n,k) of distinct values taken by k-th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 9, 9, 1, 5, 11, 17, 20, 20, 1, 6, 15, 30, 45, 48, 48, 1, 7, 20, 50, 92, 113, 115, 115, 1, 8, 26, 77, 182, 262, 283, 286, 286, 1, 9, 32, 113, 342, 591, 691, 717, 719, 719, 1, 10, 39, 156, 601, 1263, 1681, 1815, 1838, 1842, 1842
Offset: 1

Views

Author

Alois P. Heinz, Sep 05 2012

Keywords

Comments

T(n,k) <= A000081(n) because there are only A000081(n) different functions that can be represented with n x's.
It is not true that T(n,n) = T(n,n-1) for all n>1: T(13,13) - T(13,12) = 12486 - 12485 = 1.
Conjecture: T(n,n) = A000081(n) for n>=1. It would be nice to have a proof (or a disproof if the conjecture is wrong).
From Bradley Klee, Jun 01 2015 (Start):
I made a descendant graph (Plot 1) that shows how each derivative relates to the next. In this picture the number of nodes in row k gives the value T(n,k). You can see at n=6 collisions begin to occur, and at n=7 the situation is even worse. I then computed a new triangle with collisions removed (Plot 2) and values:
1
1 1
1 2 2
1 3 4 4
1 4 7 9 9
1 5 11 88 20 20
1 6 16 34 46 48 48
I suspect that Plot 2 will admit a recursive construction more readily than the graphs with collisions. You can already see that each graph "n-1" is a subgraph of graph "n" and that the remainder of graph "n" is similar to graph "n-1" with additional branches. (End)

Examples

			For n = 4 there are A000108(3) = 5 possible parenthesizations of x^x^x^x: [x^(x^(x^x)), x^((x^x)^x), (x^(x^x))^x, (x^x)^(x^x), ((x^x)^x)^x]. The first, second, third, fourth derivatives at x=1 are [1,1,1,1,1], [2,2,4,4,6], [9,15,18,18,27], [56,80,100,100,156] => row 4 = [1,3,4,4].
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  4,  4;
  1, 4,  7,  9,  9;
  1, 5, 11, 17, 20,  20;
  1, 6, 15, 30, 45,  48,  48;
  1, 7, 20, 50, 92, 113, 115, 115;
  ...

Links

Crossrefs

Columns k=1-10 give: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.
Main diagonal gives (conjectured): A000081.
Cf. A000108, A215703.

Programs

  • Maple
    with(combinat):
F:= proc(n) F(n):=`if`(n<2, [(x+1)$n], map(h->(x+1)^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [(x+1)^n],
     `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
      w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
    end:
T:= proc(n) local i, l;
      l:= map(f->[seq(i!*coeff(series(f, x, n+1), x, i), i=1..n)], F(n));
      seq(nops({map(x->x[i], l)[]}), i=1..n)
    end:
seq(T(n), n=1..10);
  • Mathematica
    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}]];
T[n_] := T[n] = If[n==1, {x}, x^#& /@ g[n-1, n-1]];
T[n_, k_] := Union[k! (SeriesCoefficient[#, {x, 0, k}]& /@ (T[n] /. x -> x+1))] // Length;
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)
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