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.

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

Author

Alois P. Heinz, Nov 12 2017

Keywords

• sign
• tabl

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

• 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

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).