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

1,13

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Eric Weisstein's World of Mathematics, Power Tower

Wikipedia, Knuth's up-arrow notation

Wikipedia, Tetration

Formula

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).

Example

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, ...

Maple

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);

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]]/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 *)

Crossrefs

Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110.

Main diagonal gives A136461(n-1).

Cf. A277537, A295027.

Sequence in context: A293202 A280265 A292795 * A294201 A079618 A151844

Adjacent sequences: A295025 A295026 A295027 * A295029 A295030 A295031

Keyword

sign,tabl

Author

Alois P. Heinz, Nov 12 2017

Status

approved