0,13

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Eric Weisstein's World of Mathematics, Power Tower

Wikipedia, Knuth's up-arrow notation

Wikipedia, Tetration

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.

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, ...

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);

# Alternative:

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);

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. *)

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.

Sequence in context: A035208 A025881 A039804 * A323179 A320508 A164925

Adjacent sequences: A277534 A277535 A277536 * A277538 A277539 A277540

sign,tabl

Alois P. Heinz, Oct 19 2016

approved