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%I #33 Oct 18 2018 16:18:44
%S 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,
%T 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,
%U 480,2934,6524,944,0,0,1,1,2,9,56,480,4374,26054,45016,-5112,0,0
%N 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.
%H Alois P. Heinz, <a href="/A277537/b277537.txt">Antidiagonals n = 0..140, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A(n,k) = [(d/dx)^n x^^k]_{x=1}.
%F E.g.f. of column k: (x+1)^^k.
%F A(n,k) = Sum_{i=0..min(n,k)} A277536(n,i).
%F A(n,k) = n * A295028(n,k) for n,k > 0.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 0, 2, 2, 2, 2, 2, 2, ...
%e 0, 0, 3, 9, 9, 9, 9, 9, ...
%e 0, 0, 8, 32, 56, 56, 56, 56, ...
%e 0, 0, 10, 180, 360, 480, 480, 480, ...
%e 0, 0, 54, 954, 2934, 4374, 5094, 5094, ...
%e 0, 0, -42, 6524, 26054, 47894, 60494, 65534, ...
%p f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A:= (n, k)-> n!*coeff(series(f(k), x, n+1), x, n):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%p # second Maple program:
%p b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
%p -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
%p (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
%p end:
%p A:= (n, k)-> b(n, min(k, n)):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t 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. *)
%Y Columns k=0..10 give A000007, A019590(n+1), A005727, A179230, A179405, A179505, A211205, A277538, A277539, A277540, A277541.
%Y Rows n=0..1 give A000012, A057427.
%Y Main diagonal gives A033917.
%Y Cf. A215703, A277536, A295028.
%K sign,tabl
%O 0,13
%A _Alois P. Heinz_, Oct 19 2016
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