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%I #25 Jan 08 2021 15:51:32
%S 1,0,2,0,3,3,0,8,12,4,0,10,85,30,5,0,54,450,330,60,6,0,-42,3283,3255,
%T 910,105,7,0,944,22036,37352,12740,2072,168,8,0,-5112,182628,441756,
%U 200781,37800,4158,252,9,0,47160,1488240,5765540,3282300,747390,94500,7620,360,10
%N T(n,k) is 1/(k-1)! times the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A298605/b298605.txt">Rows 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 T(n,k) = n!/(k-1)! * [x^n] ((x+1)^^k - (x+1)^^(k-1)).
%F T(n,k) = 1/(k-1)! * [(d/dx)^n (x^^k - x^^(k-1))]_{x=1}.
%F T(n,k) = 1/(k-1)! * A277536(n,k).
%F T(n,k) = n/(k-1)! * A295027(n,k).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 2;
%e 0, 3, 3;
%e 0, 8, 12, 4;
%e 0, 10, 85, 30, 5;
%e 0, 54, 450, 330, 60, 6;
%e 0, -42, 3283, 3255, 910, 105, 7;
%e 0, 944, 22036, 37352, 12740, 2072, 168, 8;
%e 0, -5112, 182628, 441756, 200781, 37800, 4158, 252, 9;
%e 0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10;
%e ...
%p f:= proc(n) option remember; `if`(n<0, 0,
%p `if`(n=0, 1, (x+1)^f(n-1)))
%p end:
%p T:= (n, k)-> n!/(k-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
%p seq(seq(T(n, k), k=1..n), n=1..10);
%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 T:= (n, k)-> (b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))))/(k-1)!:
%p seq(seq(T(n, k), k=1..n), n=1..10);
%t f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
%t T[n_, k_] := n!/(k - 1)!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
%t Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten
%t (* Second program: *)
%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 T[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k-1, n]]])/(k-1)!;
%t Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 03 2018, from Maple *)
%Y Columns k=1-2 give: A063524, A005727 (for n>1).
%Y Main diagonal gives A000027.
%Y Cf. A277536, A295027.
%K sign,tabl
%O 1,3
%A _Alois P. Heinz_, Jan 22 2018
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