OFFSET
0,13
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
FORMULA
EXAMPLE
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, ...
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. *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Alois P. Heinz, Oct 19 2016
STATUS
approved