A211192 -id:A211192 - cp's OEIS frontend

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.

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

Alois P. Heinz, Aug 21 2012

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  |

			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, ...

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

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.

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 *)

A222379 Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways giving result f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 6, 19, 38, 107, 247, 668, 1666, 4468, 11603, 31210, 83044, 224893, 607658, 1657966, 4528193, 12441364, 34254321, 94696165, 262389581, 729258392, 2031264865, 5671570468, 15867219821, 44480785907, 124913622052, 351393746745, 990048748684

Offset: 0

Alois P. Heinz, Feb 17 2013

nonn

A000081(n) distinct functions are representable as x -> x^x^...^x with n 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(n-1) for n>0.

			There are A000081(4) = 4 functions f representable as x -> x^x^...^x with 4 x's and parentheses inserted in all possible ways: ((x^x)^x)^x, (x^x)^(x^x) == (x^(x^x))^x, x^((x^x)^x), x^(x^(x^x)).  Only x^((x^x)^x) evaluates to 0 at x=0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0. Thus a(4) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's
Wikipedia, Zero to the power of zero

Cf. A000081, A000108, A055113, A087803, A211192, A215703, A222380, A306710.

g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
      v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
      binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
    end:
a:= n-> g(n-1$2)[2]:
seq(a(n), n=0..40);

f[l_, h_] := {Binomial[l[[2]] + l[[1]] + j - 1, j]*(h[[1]] + h[[2]]), Binomial[l[[1]] + j - 1, j]*h[[2]]};
g[n_, i_] := g[n, i] = If[n == 0, {0, 1}, If[i < 1, {0, 0}, Function[v, {v[[1]] - v[[2]], v[[2]]}][Sum[f[g[i - 1, i - 1], g[n - i*j, i - 1]], {j, 0, Quotient[n, i]}]]]];
a[n_] := g[n - 1, n - 1][[2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2019, after Alois P. Heinz *)

A222380(n) + a(n) = A000081(n).
A222380(n) - a(n) = A211192(n).
a(n) = Sum_{i=A087803(n-1)+1..A087803(n)} (1-A306710(i)).

A222380 Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways giving result f(0)=1, with conventions that 0^0=1^0=1^1=1, 0^1=0.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 14, 29, 77, 179, 472, 1174, 3100, 8018, 21370, 56601, 152337, 409954, 1113501, 3030710, 8298035, 22780468, 62800860, 173586690, 481335403, 1337916253, 3728371645, 10412163861, 29139846448, 81705768401, 229513225545, 645777766253

Offset: 0

Alois P. Heinz, Feb 17 2013

nonn

			There are A000081(4) = 4 functions f representable as x -> x^x^...^x with 4 x's and parentheses inserted in all possible ways: ((x^x)^x)^x, (x^x)^(x^x) == (x^(x^x))^x, x^((x^x)^x), x^(x^(x^x)).  Only x^((x^x)^x) evaluates to 0 at x=0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0.  Three functions evaluate to 1 at x=0: ((0^0)^0)^0 = (1^0)^0 = 1^0 = 1, (0^0)^(0^0) = 1^1 = 1, 0^(0^(0^0)) = 0^(0^1) = 0^0 = 1. Thus a(4) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's
Wikipedia, Zero to the power of zero

Cf. A000081, A000108, A055113, A087803, A211192, A215703, A222379, A306710.

g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
      v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
      binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
    end:
a:= n-> g(n-1$2)[1]:
seq(a(n), n=0..40);

f[l_, h_] := {Binomial[l[[2]] + l[[1]] + j - 1, j]*(h[[1]] + h[[2]]), Binomial[l[[1]] + j - 1, j]*h[[2]]};
g[n_, i_] := g[n, i] = If[n == 0, {0, 1}, If[i < 1, {0, 0}, Function[v, {v[[1]] - v[[2]], v[[2]]}][Sum[f[g[i - 1, i - 1], g[n - i*j, i - 1]], {j, 0, Quotient[n, i]}]]]];
a[n_] := g[n - 1, n - 1][[1]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2019, after Alois P. Heinz *)

a(n) + A222379(n) = A000081(n).
a(n) - A222379(n) = A211192(n).
a(n) = Sum_{i=A087803(n-1)+1..A087803(n)} A306710(i).

A306668 Difference between numbers of binary bracketings of 0^0^...^0 with n 0's giving the result 1 and those giving the result 0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

Original entry on oeis.org

0, -1, 1, 0, 3, 4, 20, 50, 189, 588, 2100, 7116, 25344, 89298, 321178, 1156298, 4206059, 15356796, 56424836, 208137800, 771229684, 2867771004, 10700980956, 40050890172, 150328400292, 565699287186, 2133889856550, 8067040670100, 30559571239890, 115986196679730

Offset: 0

Alois P. Heinz, Mar 04 2019

sign

The total number of binary bracketings of 0^0^...^0 with n 0's is A000108(n-1) for n > 0.

			There are A000108(3) = 5 binary bracketings of 0^0^0^0: ((0^0)^0)^0, (0^0)^(0^0), (0^(0^0))^0, 0^((0^0)^0), 0^(0^(0^0)). Only 0^((0^0)^0) evaluates to 0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0. The four other bracketings evaluate to 1. Thus a(4) = 4-1 = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1670
Eric Weisstein's World of Mathematics, Binary Bracketing
Wikipedia, Zero to the power of zero

Cf. A000079, A000108, A055113, A111160, A211192.

b:= proc(n) option remember; `if`(n<2, [n, 0], add(((f, g)-> [f[1]*g[2],
      f[1]*g[1] +f[2]*g[1] +f[2]*g[2]])(b(i), b(n-i)), i=1..n-1))
    end:
a:= n-> (v-> v[2]-v[1])(b(n)):
seq(a(n), n=0..29);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, -n, ((35*n^3-147*n^2+220*n-120)*
      a(n-1)+18*(n-2)*(5*n-6)*(2*n-5)*a(n-2))/((2*(5*n-11))*(2*n-1)*n))
    end:
seq(a(n), n=0..29);

a[n_] := a[n] = If[n<2, -n, ((35n^3 - 147n^2 + 220n - 120) a[n-1] + 18(n-2) (5n - 6)(2n - 5) a[n-2])/((2(5n - 11))(2n - 1)n)];
a /@ Range[0, 29] (* Jean-François Alcover, Apr 02 2021, after 2nd Maple program *)

a(n) = A111160(n-1) - A055113(n) for n > 0.

a(n) is odd <=> n in { A000079 }.

A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

Original entry on oeis.org

0, 3, 4, 6, 7, 9, 11, 12, 13, 19, 20, 21, 24, 26, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 66, 69, 70, 72, 79, 80, 81, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 97, 100, 101, 102, 103, 105, 106, 107

Offset: 1

Vladimir Reshetnikov, Nov 06 2015

nonn

Complement of A263831.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000081, A211192, A263831.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k-m, t[#] # &] t[m]/(k-1), {m, k-1}]; Select[Range[0, 107], EvenQ@t[#] &]] (* after Alois P. Heinz *)

A263831 Values of k such that the number of rooted trees with k nodes (A000081(k)) is odd.

Original entry on oeis.org

1, 2, 5, 8, 10, 14, 15, 16, 17, 18, 22, 23, 25, 27, 28, 30, 33, 35, 38, 40, 42, 49, 50, 56, 61, 64, 65, 67, 68, 71, 73, 74, 75, 76, 77, 78, 82, 84, 87, 93, 98, 99, 104, 108, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 127, 128, 135, 137, 138, 139

Offset: 1

Vladimir Reshetnikov, Nov 03 2015

nonn

Complement of A258592.

Cf. A000081, A211192, A258592.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k-m, t[#] # &] t[m]/(k-1), {m, k-1}]; Select[Range[140], OddQ@t[#] &]] (* after Alois P. Heinz *)

cp's OEIS Frontend

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.

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A222379 Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways giving result f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

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A222380 Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways giving result f(0)=1, with conventions that 0^0=1^0=1^1=1, 0^1=0.

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A306668 Difference between numbers of binary bracketings of 0^0^...^0 with n 0's giving the result 1 and those giving the result 0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

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A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

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A263831 Values of k such that the number of rooted trees with k nodes (A000081(k)) is odd.

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