A295027 -id:A295027 - cp's OEIS frontend

A104150 Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.

0, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 0

Miklos Kristof, Mar 08 2005

E.g.f.: Sum_{n>=1} (n-1)!*x^n/n! = Sum_{n>=1} x^n/n.
The shift law of the e.g.f.: if Sum_{n>=0} a(n)*x^n/n! = f(x), then Sum_{n>=0} a(n+1)*x^n/n! = d/dx f(x) and Sum_{n>=1} a(n-1)*x^n/n! = Integral f(x) dx.
The e.g.f. of A000142 (= n!) is 1/(1-x), so the e.g.f. of a(n)=(n-1)! is integral 1/(1-x) = -log(1-x).

A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. N. Gladkovskii, Analysis Of The Continued Fractions (in Russian), see p.79 (5.1.21)

Cf. A000142.
Column k=1 of A285849.
Main diagonal of A295027 (for n > 0).

[0] cat [Factorial(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 25 2012

Join[{0,1},Range[20]!] (* Harvey P. Dale, Dec 09 2013 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace(-log(1-x)))) \\ G. C. Greubel, May 15 2018

[stirling_number1(n,1) for n in range(0, 22)] # Zerinvary Lajos, May 16 2009

E.g.f. -log(1-x) = x + x^2/2 + x^3/3 + ... + x^n/n + ...
G.f.: x+x^2/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); G(0) = W(1,1;-x)/W(1,2;-x), W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)], x-> -x;  (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012
E.g.f.: (-x + 5*x^2/2 - 11*x^3/6 + x^4/4 + x^5/(W(0)-x)/4)/(x-1)^3 where W(k)= (x + 1)*k + x + 5 - x*(k+2)*(k+5)/W(k+1); see [S. N. Gladkovskii, p. 79 (5.1.21)]; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 15 2012
G.f.: A(x) = Integral_{t>=0} x*exp(-t)/(1-x*t) dt = x/G(0) where G(k) = 1 - x*(k+1)/(1 - x*(k+1)/G(k+1)); (continued fraction due to L. Euler and E. N. Laguerre). - Sergei N. Gladkovskii, Dec 24 2012
G.f.: x + x/Q(0), where Q(k)= 1/x - (2*k+2) - (k+2)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013
G.f.: x/Q(0), where Q(k) = 1 - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
G.f.: x*G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 07 2013

A295028 A(n,k) is (1/n) times 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>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 8, 2, 0, 1, 1, 3, 14, 36, 9, 0, 1, 1, 3, 14, 72, 159, -6, 0, 1, 1, 3, 14, 96, 489, 932, 118, 0, 1, 1, 3, 14, 96, 729, 3722, 5627, -568, 0, 1, 1, 3, 14, 96, 849, 6842, 33641, 40016, 4716, 0

Offset: 1

Alois P. Heinz, Nov 12 2017

			Square array A(n,k) begins:
  1,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    3,     3,     3,      3,      3,      3, ...
  0,   2,    8,    14,    14,     14,     14,     14, ...
  0,   2,   36,    72,    96,     96,     96,     96, ...
  0,   9,  159,   489,   729,    849,    849,    849, ...
  0,  -6,  932,  3722,  6842,   8642,   9362,   9362, ...
  0, 118, 5627, 33641, 71861, 102941, 118061, 123101, ...

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

Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110.
Main diagonal gives A136461(n-1).
Cf. A277537, A295027.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> (n-1)!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..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))/n:
seq(seq(A(n, 1+d-n), n=1..d), d=1..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]]/n;
Table[A[n, 1 + d - n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 25 2018, translated from 2nd Maple program *)

A(n,k) = 1/n * [(d/dx)^n x^^k]_{x=1}.
A(n,k) = (n-1)! * [x^n] (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A295027(n,i).
A(n,k) = 1/n * A277537(n,k).

A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 3, 6, 0, 0, 8, 24, 24, 0, 0, 10, 170, 180, 120, 0, 0, 54, 900, 1980, 1440, 720, 0, 0, -42, 6566, 19530, 21840, 12600, 5040, 0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320, 0, 0, -5112, 365256, 2650536, 4818744, 4536000, 2993760, 1270080, 362880

Offset: 0

Alois P. Heinz, Oct 19 2016

T(n,k) is defined for all n,k >= 0. 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, 0,   2;
  0, 0,   3,     6;
  0, 0,   8,    24,     24;
  0, 0,  10,   170,    180,    120;
  0, 0,  54,   900,   1980,   1440,    720;
  0, 0, -42,  6566,  19530,  21840,  12600,   5040;
  0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320;
  ...

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

Columns k=0-2 give: A000007, A063524, A005727 (for n>1).
Main diagonal gives A000142.
Row sums give A033917.
T(n+1,n)/3 gives A005990.
T(2n,n) gives A290023.
Cf. A277537, A295027.

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

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

E.g.f. of column k>0: (x+1)^^k - (x+1)^^(k-1), e.g.f. of column k=0: 1.
T(n,k) = [(d/dx)^n (x^^k - x^^(k-1))]_{x=1} for k>0, T(n,0) = A000007(n).
T(n,k) = A277537(n,k) - A277537(n,k-1) for k>0, T(n,0) = A000007(n).
T(n,k) = n * A295027(n,k) for n,k > 0.

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

A298605 T(n,k) is 1/(k-1)! 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, 2, 0, 3, 3, 0, 8, 12, 4, 0, 10, 85, 30, 5, 0, 54, 450, 330, 60, 6, 0, -42, 3283, 3255, 910, 105, 7, 0, 944, 22036, 37352, 12740, 2072, 168, 8, 0, -5112, 182628, 441756, 200781, 37800, 4158, 252, 9, 0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10

Offset: 1

Alois P. Heinz, Jan 22 2018

			Triangle T(n,k) begins:
  1;
  0,     2;
  0,     3,       3;
  0,     8,      12,       4;
  0,    10,      85,      30,       5;
  0,    54,     450,     330,      60,      6;
  0,   -42,    3283,    3255,     910,    105,     7;
  0,   944,   22036,   37352,   12740,   2072,   168,    8;
  0, -5112,  182628,  441756,  200781,  37800,  4158,  252,   9;
  0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10;
  ...

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

Columns k=1-2 give: A063524, A005727 (for n>1).
Main diagonal gives A000027.
Cf. A277536, A295027.

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
T:= (n, k)-> n!/(k-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10);
# 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))))/(k-1)!:
seq(seq(T(n, k), k=1..n), n=1..10);

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := n!/(k - 1)!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
Table[T[n, k], {n, 1, 10}, { 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]]])/(k-1)!;
Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

T(n,k) = n!/(k-1)! * [x^n] ((x+1)^^k - (x+1)^^(k-1)).
T(n,k) = 1/(k-1)! * [(d/dx)^n (x^^k - x^^(k-1))]_{x=1}.
T(n,k) = 1/(k-1)! * A277536(n,k).
T(n,k) = n/(k-1)! * A295027(n,k).

cp's OEIS Frontend

A104150 Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.

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A295028 A(n,k) is (1/n) times 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>=1, k>=1, read by antidiagonals.

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A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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

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

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A298605 T(n,k) is 1/(k-1)! 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.

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