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%I #18 May 25 2018 04:24:18
%S 1,1,0,1,1,0,1,1,1,0,1,1,3,2,0,1,1,3,8,2,0,1,1,3,14,36,9,0,1,1,3,14,
%T 72,159,-6,0,1,1,3,14,96,489,932,118,0,1,1,3,14,96,729,3722,5627,-568,
%U 0,1,1,3,14,96,849,6842,33641,40016,4716,0
%N A(n,k) is (1/n) times 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>=1, k>=1, read by antidiagonals.
%H Alois P. Heinz, <a href="/A295028/b295028.txt">Antidiagonals n = 1..141, 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) = 1/n * [(d/dx)^n x^^k]_{x=1}.
%F A(n,k) = (n-1)! * [x^n] (x+1)^^k.
%F A(n,k) = Sum_{i=0..min(n,k)} A295027(n,i).
%F A(n,k) = 1/n * A277537(n,k).
%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, 1, 3, 3, 3, 3, 3, 3, ...
%e 0, 2, 8, 14, 14, 14, 14, 14, ...
%e 0, 2, 36, 72, 96, 96, 96, 96, ...
%e 0, 9, 159, 489, 729, 849, 849, 849, ...
%e 0, -6, 932, 3722, 6842, 8642, 9362, 9362, ...
%e 0, 118, 5627, 33641, 71861, 102941, 118061, 123101, ...
%p f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A:= (n, k)-> (n-1)!*coeff(series(f(k), x, n+1), x, n):
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..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))/n:
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..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}]]];
%t A[n_, k_] := b[n, Min[k, n]]/n;
%t Table[A[n, 1 + d - n], {d, 1, 14}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, May 25 2018, translated from 2nd Maple program *)
%Y Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110.
%Y Main diagonal gives A136461(n-1).
%Y Cf. A277537, A295027.
%K sign,tabl
%O 1,13
%A _Alois P. Heinz_, Nov 12 2017
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