A277537 - OEIS

A277537

A(n,k) is 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>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 9, 8, 0, 0, 1, 1, 2, 9, 32, 10, 0, 0, 1, 1, 2, 9, 56, 180, 54, 0, 0, 1, 1, 2, 9, 56, 360, 954, -42, 0, 0, 1, 1, 2, 9, 56, 480, 2934, 6524, 944, 0, 0, 1, 1, 2, 9, 56, 480, 4374, 26054, 45016, -5112, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

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

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.

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

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

KEYWORD

sign,tabl

AUTHOR

Alois P. Heinz, Oct 19 2016

STATUS

approved