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 *)

A216349 Triangle T(n,k) in which n-th row lists the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

Original entry on oeis.org

1, 2, 12, 9, 156, 100, 80, 56, 3160, 1880, 1180, 1420, 950, 1360, 890, 660, 480, 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, 14988, 10848, 34974, 21474, 13314, 15114, 10974, 13014, 8874, 6534, 5094, 3218628, 1806476, 1021552, 588756, 1189132
Offset: 1

Views

Author

Alois P. Heinz, Sep 04 2012

Keywords

Comments

The ordering of the functions is the same as in A215703 and is defined by the algorithm below.

Examples

			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.
Triangle T(n,k) begins:
:     1;
:     2;
:    12,     9;
:   156,   100,    80,    56;
:  3160,  1880,  1180,  1420,   950,  1360,   890,   660,   480;
: 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, ...

Links

Crossrefs

First column gives: A216351.
Last elements of rows give: A033917.
A version with sorted row elements is: A216350.
Rows sums give: A216281.
Cf. A000081, A215703.

Programs

  • Maple
    with(combinat):
F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^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:= n-> map(f-> n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), F(n))[]:
seq(T(n), n=1..7);

A216350 Triangle T(n,k) in which n-th row lists in increasing order the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

Original entry on oeis.org

1, 2, 9, 12, 56, 80, 100, 156, 480, 660, 890, 950, 1180, 1360, 1420, 1880, 3160, 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, 20268, 21474, 22008, 24042, 29682, 31968, 34974, 35382, 50496, 87990, 65534, 78134, 102494, 131684, 141974
Offset: 1

Views

Author

Alois P. Heinz, Sep 04 2012

Keywords

Examples

			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 => 4th row = [56, 80, 100, 156].
Triangle T(n,k) begins:
:    1;
:    2;
:    9,   12;
:   56,   80,  100,   156;
:  480,  660,  890,   950,  1180,  1360,  1420,  1880,  3160;
: 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, ...

Links

Crossrefs

First column gives: A033917.
Last elements of rows give: A216351.
A version with different ordering of row elements is: A216349.
Rows sums give: A216281.
Cf. A000081, A215703.

Programs

  • Maple
    with(combinat):
F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^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:= n-> sort(map(f-> n!*coeff(series(subs(x=x+1, f)
                 , x, n+1), x, n), F(n)))[]:
seq(T(n), n=1..7);
Showing 1-3 of 3 results.

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