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.

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

Author

Alois P. Heinz, Jan 22 2018

Keywords

• sign
• tabl

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

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

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).