A136461 -id:A136461 - cp's OEIS frontend

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.

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).

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).

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula