A179405 -id:A179405 - 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 *)

A215524 n-th derivative of (x^x)^x at x=1.

Original entry on oeis.org

1, 1, 4, 12, 52, 240, 1188, 6804, 38960, 253296, 1654560, 11816640, 85816608, 668005728, 5240582592, 44667645120, 365989405440, 3494595006720, 28075694694912, 325862541872640, 2101211758356480, 39605981661066240, 48568198208302080, 7549838510211486720, -66667098077331572736

Offset: 0

Alois P. Heinz, Aug 14 2012

sign

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=3 of A215703.

Cf. A005727, A179230, A179405, A215522.

a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^x ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 24; CoefficientList[((x+1)^(x+1))^(x+1) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: ((x+1)^(x+1))^(x+1).

A277537 A(n,k) is the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 9, 8, 0, 0, 1, 1, 2, 9, 32, 10, 0, 0, 1, 1, 2, 9, 56, 180, 54, 0, 0, 1, 1, 2, 9, 56, 360, 954, -42, 0, 0, 1, 1, 2, 9, 56, 480, 2934, 6524, 944, 0, 0, 1, 1, 2, 9, 56, 480, 4374, 26054, 45016, -5112, 0, 0

Offset: 0

Alois P. Heinz, Oct 19 2016

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,     1,     1,     1, ...
  0, 1,   1,    1,     1,     1,     1,     1, ...
  0, 0,   2,    2,     2,     2,     2,     2, ...
  0, 0,   3,    9,     9,     9,     9,     9, ...
  0, 0,   8,   32,    56,    56,    56,    56, ...
  0, 0,  10,  180,   360,   480,   480,   480, ...
  0, 0,  54,  954,  2934,  4374,  5094,  5094, ...
  0, 0, -42, 6524, 26054, 47894, 60494, 65534, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=0..10 give A000007, A019590(n+1), A005727, A179230, A179405, A179505, A211205, A277538, A277539, A277540, A277541.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A033917.
Cf. A215703, A277536, A295028.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> n!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
      -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
      (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
    end:
A:= (n, k)-> b(n, min(k, n)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_] := b[n, k] = If[n==0, 1, If[k==0, 0, -Sum[Binomial[n-1, j]*b[j, k]*Sum[Binomial[n-j, i]*(-1)^i*b[n-j-i, k-1]*(i-1)!, {i, 1, n-j}], {j, 0, n-1}]]]; A[n_, k_] := b[n, Min[k, n]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 14 2017, adapted from 2nd Maple prog. *)

A(n,k) = [(d/dx)^n x^^k]_{x=1}.
E.g.f. of column k: (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A277536(n,i).
A(n,k) = n * A295028(n,k) for n,k > 0.

A215522 n-th derivative of (x^x)^(x^x) at x=1.

Original entry on oeis.org

1, 1, 4, 18, 100, 650, 4908, 41090, 382520, 3863808, 42409440, 497972112, 6259762320, 83343114504, 1175904241848, 17442325040520, 272149555445760, 4438451554802880, 75714874759039104, 1343817666163911168, 24837691533530152320, 475811860099666527360

Offset: 0

Alois P. Heinz, Aug 14 2012

sign

Also n-th derivative of (x^(x^x))^x = x^(x^x*x) at x=1.

First term < 0: a(65).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=6 of A215703.

Cf. A005727, A179230, A179405, A215524.

a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^(x^x) ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 21; CoefficientList[(x+1)^((x+1)^(x+2)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: (x+1)^((x+1)^(x+2)).

A179505 n-th derivative of x^(x^(x^(x^x))) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 4374, 47894, 574888, 7829424, 116392080, 1901059512, 33564909432, 639562529424, 13047133134840, 283976169754440, 6563364026374464, 160538113862231808, 4141949353327046592, 112396373034208003008, 3199752121483607518080

Offset: 0

Robert G. Wilson v, Jul 17 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A179230, A179405, A295105.
Column k=17 of A215703.
Column k=5 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^x))) ), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 21 2012

f[n_] := D[x^(x^(x^(x^x))), {x, n}] /. x -> 1; Array[f, 16, 0]
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^((x+1)^((x+1)^(x+1)))). - Alois P. Heinz, Aug 21 2012

a(16)-a(20) from Alois P. Heinz, Aug 21 2012

A215704 n-th derivative of ((x^x)^x)^x at x=1.

Original entry on oeis.org

1, 1, 6, 27, 156, 1110, 8322, 70098, 646272, 6333336, 66712680, 745731360, 8780828328, 108873486072, 1413807287760, 19157627737080, 270460073295360, 3965693824244160, 60266513065134528, 947644484349584448, 15389579447794454400, 257702782790624613120

Offset: 0

Alois P. Heinz, Aug 21 2012

sign

Also n-th derivative of x^(x^3) at x=1.

First term < 0: a(57).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=5 of A215703.

Cf. A005727, A179230, A179405, A215524.

a:= n-> n!*coeff(series(subs(x=x+1, ((x^x)^x)^x ), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: (x+1)^((x+1)^3).

A211205 n-th derivative of x^(x^(x^(x^(x^x)))) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 60494, 823528, 12365424, 206078880, 3745686912, 74083090872, 1579529362944, 36165466533000, 884104045301640, 22992315801392064, 633547543117707648, 18439576158792912192, 565162707747635408448, 18194047307015185486080

Offset: 0

Alois P. Heinz and Robert G. Wilson v, Feb 04 2013

nonn

Robert G. Wilson v and Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A008405, A176118, A179230, A179405, A179505, A215522, A215524, A215629, A215643, A215691, A215704, A215705, A215706, A215707, A215708, A215709, A215710, A215522, A295106.
Column k=37 of A215703.
Column k=6 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^(x^x))))), x, n+1), x, n):
seq(a(n), n=0..20);

NestList[ Factor[ D[#1, x]] &, x^x^x^x^x^x, 9] /. (x -> 1) (* or quicker *)
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x]

E.g.f.: (x+1)^((x+1)^((x+1)^((x+1)^((x+1)^(x+1))))).

A215643 n-th derivative of x^((x^(x^x))^x) at x=1.

Original entry on oeis.org

1, 1, 2, 15, 104, 890, 8814, 100660, 1288048, 18337680, 286674960, 4882660464, 89880715704, 1777384045944, 37552294300416, 843830334815640, 20086549955304384, 504750167170162944, 13348550475903813120, 370499740676381737728, 10766442934111876381440

Offset: 0

Alois P. Heinz, Aug 18 2012

nonn

Also n-th derivative of x^((x^x)^(x^x)) = x^(x^(x^x*x)) at x=1.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=15 of A215703.

Cf. A005727, A179230, A179405, A215522, A215524.

a:= n-> n!*coeff(series(subs(x=x+1, x^((x^(x^x))^x) ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 20;
CoefficientList[(x+1)^(((x+1)^((x+1)^(x+1)))^(x+1)) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: (x+1)^(((x+1)^((x+1)^(x+1)))^(x+1)).

A215629 n-th derivative of x^(x^((x^x)^x)) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 80, 660, 6714, 77084, 1005640, 14572944, 233086920, 4066783512, 76906345944, 1566049091568, 34153725715368, 793996577407560, 19595885746343808, 511550462381982528, 14080034085212120256, 407434430977558009344, 12363449947108075756800

Offset: 0

Alois P. Heinz, Aug 18 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=16 of A215703.

Cf. A005727, A179230, A179405, A215522, A215524.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^((x^x)^x)) ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 20;
CoefficientList[(x+1)^((x+1)^(((x+1)^(x+1))^(x+1))) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f: (x+1)^((x+1)^(((x+1)^(x+1))^(x+1))).

A215691 n-th derivative of (x^x)^(x^(x^x)) at x=1.

Original entry on oeis.org

1, 1, 4, 18, 124, 950, 8688, 89600, 1038392, 13309272, 186471480, 2837173152, 46466835072, 815532508440, 15246845864040, 302533865599800, 6344720827608384, 140208886623418752, 3254819745378435264, 79172189409906466560, 2013138139856523598080

Offset: 0

Alois P. Heinz, Aug 20 2012

nonn

Also n-th derivative of (x^(x^(x^x)))^x = x^(x^(x^x)*x) at x=1.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=13 of A215703.

Cf. A005727, A179230, A179405, A215522, A215524, A215643.

a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^(x^(x^x))), x, n+1), x, n):
seq(a(n), n=0..30);

m = 20;
CoefficientList[(x+1)^((x+1)^((x+1)^(x+1)+1)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: (x+1)^((x+1)^((x+1)^(x+1)+1)).

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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A215524 n-th derivative of (x^x)^x at x=1.

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A277537 A(n,k) is the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A215522 n-th derivative of (x^x)^(x^x) at x=1.

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A179505 n-th derivative of x^(x^(x^(x^x))) at x=1.

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A215704 n-th derivative of ((x^x)^x)^x at x=1.

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A211205 n-th derivative of x^(x^(x^(x^(x^x)))) at x=1.

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A215643 n-th derivative of x^((x^(x^x))^x) at x=1.

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