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.

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

Author

Alois P. Heinz, Oct 19 2016

Keywords

• sign
• tabl

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

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

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.