OFFSET
1,6
COMMENTS
T(n,k) is defined for all n,k >= 1. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
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, 1;
0, 1, 2;
0, 2, 6, 6;
0, 2, 34, 36, 24;
0, 9, 150, 330, 240, 120;
0, -6, 938, 2790, 3120, 1800, 720;
0, 118, 5509, 28014, 38220, 31080, 15120, 5040;
0, -568, 40584, 294504, 535416, 504000, 332640, 141120, 40320;
...
MAPLE
f:= proc(n) option remember; `if`(n<0, 0,
`if`(n=0, 1, (x+1)^f(n-1)))
end:
T:= (n, k)-> (n-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..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))))/n:
seq(seq(T(n, k), k=1..n), n=1..12);
MATHEMATICA
f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := (n - 1)!*SeriesCoefficient[f[k] - f[k - 1], {x, 0, n}];
Table[T[n, k], {n, 1, 12}, {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]]])/n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 28 2018, from Maple *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Alois P. Heinz, Nov 12 2017
STATUS
approved