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.

Showing 1-5 of 5 results.

A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 65534, 984808, 16992144, 330667680, 7170714672, 171438170232, 4480972742064, 127115833240200, 3889913061111240, 127729720697035584, 4479821940873927168, 167143865005981109952, 6610411989494027218368, 276242547290322145178880
Offset: 0

Views

Author

Patrick D McLean

Keywords

Comments

a(n) is the n-th derivative of x^(x^...(x^(x^x))) with n x's evaluated at x=1. - Alois P. Heinz, Oct 14 2016

Links

Crossrefs

Cf. A052880, A215703, A216349, A216350.
Row sums of A277536.
Main diagonal of A277537.

Programs

  • Maple
    a:= n-> add(Stirling1(n, k)*(k+1)^(k-1), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 31 2012
  • Mathematica
    mx = 20; Table[ i! SeriesCoefficient[ InverseSeries[ Series[ y^(1/y), {y, 1, mx}]], i], {i, 0, n}] (* modified by Robert G. Wilson v, Feb 03 2013 *)
CoefficientList[Series[-LambertW[-Log[1+x]]/Log[1+x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
  • PARI
    Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)
a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Mar 09 2013

Formula

E.g.f.: -LambertW(-log(1+x))/log(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ n^(n-1) / ( exp(n -3/2 + exp(-1)/2) * (exp(exp(-1))-1)^(n-1/2) ). - Vaclav Kotesovec, Nov 27 2012
E.g.f.: A(x) satisfies A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling1(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
a(n) = n! * [x^n] (x+1)^^n. - Alois P. Heinz, Oct 19 2016

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

Views

Author

Alois P. Heinz, Oct 19 2016

Keywords

Examples

			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, ...

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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. *)

Formula

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

Views

Author

Alois P. Heinz, Nov 12 2017

Keywords

Comments

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.

Examples

			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;
  ...

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).

A290023 a(n) is the 2n-th derivative of the difference between the n-th tetration of x (power tower of order n) and its predecessor (or 0 if n=0) at x=1.

Original entry on oeis.org

1, 0, 8, 900, 224112, 78775200, 40518181440, 28340179227360, 26078095792869120, 30544708065077606400, 44428404658605222528000, 78604530683773395984883200, 166295474965751756924207462400, 414658685362517268992110471680000, 1203746810444949373635048911870976000
Offset: 0

Views

Author

Alois P. Heinz, Nov 10 2017

Keywords

Links

Crossrefs

Cf. A277536.

Programs

  • Maple
    f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
a:= n-> (2*n)!*coeff(series(f(n)-f(n-1), x, 2*n+1), x, 2*n):
seq(a(n), n=0..15);
# 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-> b(2*n, n) -`if`(n=0, 0, b(2*n, n-1)):
seq(a(n), n=0..15);
  • Mathematica
    f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
a[n_] := (2*n)!*SeriesCoefficient[f[n] - f[n - 1], {x, 0, 2*n}];
Table[a[n], {n, 0, 15}]
(* 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}]]];
a[n_] := b[2*n, n] - If[n == 0, 0, b[2*n, n - 1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = (2n)! * [x^(2n)] (x+1)^^n - (x+1)^^(n-1) for n>0, a(0) = 1.
a(n) = [(d/dx)^(2n) (x^^n - x^^(n-1))]_{x=1} for n>0, a(0) = 1.
a(n) = A277536(2n,n).

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

Views

Author

Alois P. Heinz, Jan 22 2018

Keywords

Examples

			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;
  ...

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).
Showing 1-5 of 5 results.

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