A295028 -id:A295028 - cp's OEIS frontend

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.

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.

A295027 T(n,k) is (1/n) times the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 2, 34, 36, 24, 0, 9, 150, 330, 240, 120, 0, -6, 938, 2790, 3120, 1800, 720, 0, 118, 5509, 28014, 38220, 31080, 15120, 5040, 0, -568, 40584, 294504, 535416, 504000, 332640, 141120, 40320, 0, 4716, 297648, 3459324, 7877520, 8968680, 6804000, 3840480, 1451520, 362880

Offset: 1

Alois P. Heinz, Nov 12 2017

T(n,k) is defined for all n,k >= 1. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     6,      6;
  0,    2,    34,     36,     24;
  0,    9,   150,    330,    240,    120;
  0,   -6,   938,   2790,   3120,   1800,    720;
  0,  118,  5509,  28014,  38220,  31080,  15120,   5040;
  0, -568, 40584, 294504, 535416, 504000, 332640, 141120, 40320;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=2 gives A005168 for n>1.
Row sums give A136461(n-1).
Main diagonal gives A104150 (for n>0).
Cf. A001286, A277536, A295028.

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
T:= (n, k)-> (n-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..12);
# 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:
T:= (n, k)-> (b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))))/n:
seq(seq(T(n, k), k=1..n), n=1..12);

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := (n - 1)!*SeriesCoefficient[f[k] - f[k - 1], {x, 0, n}];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten
(* second program: *)
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}]]];
T[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k - 1, n]]])/n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 28 2018, from Maple *)

T(n,k) = (n-1)! * [x^n] ((x+1)^^k - (x+1)^^(k-1)).
T(n,k) = 1/n * [(d/dx)^n (x^^k - x^^(k-1))]_{x=1}.
T(n,k) = A295028(n,k) - A295028(n,k-1).
T(n,k) = 1/n * A277536(n,k).
T(n+1,n) = A001286(n).

A005168 n-th derivative of x^x at 1, divided by n.

Original entry on oeis.org

1, 1, 1, 2, 2, 9, -6, 118, -568, 4716, -38160, 358126, -3662088, 41073096, -500013528, 6573808200, -92840971200, 1402148010528, -22554146644416, 385014881294496, -6952611764874240, 132427188835260480, -2653529921603890560, 55802195178451990896

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..400 (first 100 terms from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane, 1986
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Cf. A005727.

Column k=2 of A295027 (for n>1), A295028.

a:= n-> (n-1)! *coeftayl(x^x, x=1, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 18 2012

Rest[(NestList[ Factor[ D[ #1, x]] &, x^x, 23] /. (x -> 1))/Range[0, 23]] (* Robert G. Wilson v, Aug 10 2010 *)

from sympy import var, diff
x = var('x')
y = x**x
l = [[y:=diff(y),y.subs(x,1)/(n+1)][1] for n in range(10)]
print(l) # Nicholas Stefan Georgescu, Mar 02 2023

One more term from Robert G. Wilson v, Aug 10 2010

A136461 Expansion of e.g.f.: A(x) = -(1 + LambertW(-log(1+x))/log(1+x))/x.

Original entry on oeis.org

1, 1, 3, 14, 96, 849, 9362, 123101, 1888016, 33066768, 651883152, 14286514186, 344690210928, 9079702374300, 259327537407416, 7983107543564724, 263518937698466304, 9285770278110061664, 347916420499685643072, 13812127364516107258944, 579183295530010157485824

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

A033917 gives the coefficients of iterated exponential function defined by y(x) = x^y(x) expanded about x=1.

Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 71 terms from Vincenzo Librandi)

Cf. A033917.
Row sums of A295027 (shifted).
Main diagonal of A295028 (shifted).

a:= n-> add(Stirling1(n+1, k)*(k+1)^(k-1), k=0..n+1)/(n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2016

CoefficientList[Series[-(1+LambertW[-Log[1+x]]/Log[1+x])/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

{a(n)=n!*polcoeff(sum(i=0,n+1,(i+1)^(i-1)*log(1+x +O(x^(n+2) ))^i/i!), n+1)}

x='x+O('x^30); Vec(serlaplace(-(1+lambertw(-log(1+x))/log(1+x))/x  )) \\ G. C. Greubel, Feb 19 2018

a(n) = A033917(n+1)/(n+1).
E.g.f.: A(x) = (1/x)*Sum_{i>=1} (i+1)^(i-1) * log(1+x)^i/i!.
a(n) ~ n^(n-1) / ( exp(n-3/2+exp(-1)/2) * (exp(exp(-1))-1)^(n+1/2) ). - Vaclav Kotesovec, Nov 27 2012

A295103 a(n) = (1/n) times the n-th derivative of the third tetration of x (power tower of order 3) x^^3 at x=1.

Original entry on oeis.org

1, 1, 3, 8, 36, 159, 932, 5627, 40016, 302364, 2510712, 22623490, 213486864, 2227719948, 23388469400, 277570328040, 3182959484736, 42530335589088, 523078873327872, 7846745537655360, 101370634558327680, 1717052148685665792, 22657314273376353408

Offset: 1

Alois P. Heinz, Nov 14 2017

sign

First term < 0: a(33) = -26329560314038014690778779463680.

Alois P. Heinz, Table of n, a(n) for n = 1..453
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=3 of A295028.

Cf. A179230.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(3), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[3], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^3]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^3.
a(n) = 1/n * A179230(n).

A295104 a(n) = (1/n) times the n-th derivative of the fourth tetration of x (power tower of order 4) x^^4 at x=1.

Original entry on oeis.org

1, 1, 3, 14, 72, 489, 3722, 33641, 334520, 3761688, 45898272, 615641806, 8863726704, 137786878644, 2279658872696, 40229212948404, 750433323448128, 14801457167223872, 306869893647304896, 6683254543551623904, 152281219079726183040, 3626445842114839589952

Offset: 1

Alois P. Heinz, Nov 14 2017

sign

First term < 0: a(329).

Alois P. Heinz, Table of n, a(n) for n = 1..452
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=4 of A295028.

Cf. A179405.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(4), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[4], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^4]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^4.
a(n) = 1/n * A179405(n).

A295105 a(n) = (1/n) times the n-th derivative of the fifth tetration of x (power tower of order 5) x^^5 at x=1.

Original entry on oeis.org

1, 1, 3, 14, 96, 729, 6842, 71861, 869936, 11639208, 172823592, 2797075786, 49197117648, 931938081060, 18931744650296, 410210251648404, 9443418462484224, 230108297407058144, 5915598580747789632, 159987606074180375904, 4539874767394840811904

Offset: 1

Alois P. Heinz, Nov 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..434
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=5 of A295028.

Cf. A179505.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(5), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[5], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^5]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^5.
a(n) = 1/n * A179505(n).

A295106 a(n) = (1/n) times the n-th derivative of the sixth tetration of x (power tower of order 6) x^^6 at x=1.

Original entry on oeis.org

1, 1, 3, 14, 96, 849, 8642, 102941, 1373936, 20607888, 340516992, 6173590906, 121502258688, 2583247609500, 58940269686776, 1437019737587004, 37267502536335744, 1024420897710717344, 29745405670928179392, 909702365350759274304, 29224500667382460549504

Offset: 1

Alois P. Heinz, Nov 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..423
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=6 of A295028.

Cf. A211205.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(6), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[6], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^6]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^6.
a(n) = 1/n * A211205(n).

A295107 a(n) = (1/n) times the n-th derivative of the seventh tetration of x (power tower of order 7) x^^7 at x=1.

Original entry on oeis.org

1, 1, 3, 14, 96, 849, 9362, 118061, 1706576, 27411888, 488133552, 9504647866, 201394553808, 4607546125740, 113271179680136, 2976610819616004, 83276079152315904, 2470817772641667104, 77492234876034762432, 2561350116102926727744, 88984716683633511515904

Offset: 1

Alois P. Heinz, Nov 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..416
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=7 of A295028.

Cf. A277538.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(7), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[7], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^7]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^7.
a(n) = 1/n * A277538(n).

A295108 a(n) = (1/n) times the n-th derivative of the eighth tetration of x (power tower of order 8) x^^8 at x=1.

Original entry on oeis.org

1, 1, 3, 14, 96, 849, 9362, 123101, 1847696, 31252368, 584145552, 11981318986, 267050704368, 6432872588700, 166461202886456, 4606491806670324, 135733988375074944, 4243153626928512224, 140252989224067186752, 4887395830953148166784, 179067423776388634331904

Offset: 1

Alois P. Heinz, Nov 14 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..412
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=8 of A295028.

Cf. A277539.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> (n-1)!*coeff(series(f(8), x, n+1), x, n):
seq(a(n), n=1..23);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := (n - 1)!*SeriesCoefficient[f[8], {x, 0, n}];
Array[a, 23] (* Jean-François Alcover, May 31 2018, from Maple *)

a(n) = 1/n * [(d/dx)^n x^^8]_{x=1}.
a(n) = (n-1)! * [x^n] (x+1)^^8.
a(n) = 1/n * A277539(n).

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A295027 T(n,k) is (1/n) times the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A005168 n-th derivative of x^x at 1, divided by n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A136461 Expansion of e.g.f.: A(x) = -(1 + LambertW(-log(1+x))/log(1+x))/x.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A295103 a(n) = (1/n) times the n-th derivative of the third tetration of x (power tower of order 3) x^^3 at x=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A295104 a(n) = (1/n) times the n-th derivative of the fourth tetration of x (power tower of order 4) x^^4 at x=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A295105 a(n) = (1/n) times the n-th derivative of the fifth tetration of x (power tower of order 5) x^^5 at x=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica