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.

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

Author

Alois P. Heinz, Oct 19 2016

Keywords

• sign
• tabl

Comments

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.

Examples

			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;
  ...
		

Links

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

Crossrefs

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.

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!*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);

Mathematica

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 *)

Formula

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.