OFFSET
1,3
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
FORMULA
EXAMPLE
Triangle T(n,k) begins:
1;
0, 2;
0, 3, 3;
0, 8, 12, 4;
0, 10, 85, 30, 5;
0, 54, 450, 330, 60, 6;
0, -42, 3283, 3255, 910, 105, 7;
0, 944, 22036, 37352, 12740, 2072, 168, 8;
0, -5112, 182628, 441756, 200781, 37800, 4158, 252, 9;
0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10;
...
MAPLE
f:= proc(n) option remember; `if`(n<0, 0,
`if`(n=0, 1, (x+1)^f(n-1)))
end:
T:= (n, k)-> n!/(k-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10);
# 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))))/(k-1)!:
seq(seq(T(n, k), k=1..n), n=1..10);
MATHEMATICA
f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := n!/(k - 1)!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
Table[T[n, k], {n, 1, 10}, { 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]]])/(k-1)!;
Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Alois P. Heinz, Jan 22 2018
STATUS
approved