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A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A277536 #35 Feb 16 2025 08:33:37
%S A277536 1,0,1,0,0,2,0,0,3,6,0,0,8,24,24,0,0,10,170,180,120,0,0,54,900,1980,
%T A277536 1440,720,0,0,-42,6566,19530,21840,12600,5040,0,0,944,44072,224112,
%U A277536 305760,248640,120960,40320,0,0,-5112,365256,2650536,4818744,4536000,2993760,1270080,362880
%N A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A277536 T(n,k) is defined for all n,k >= 0.  The triangle contains only the terms with k<=n.  T(n,k) = 0 for k>n.
%H A277536 Alois P. Heinz, <a href="/A277536/b277536.txt">Rows n = 0..140, flattened</a>
%H A277536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A277536 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A277536 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A277536 E.g.f. of column k>0: (x+1)^^k - (x+1)^^(k-1), e.g.f. of column k=0: 1.
%F A277536 T(n,k) = [(d/dx)^n (x^^k - x^^(k-1))]_{x=1} for k>0, T(n,0) = A000007(n).
%F A277536 T(n,k) = A277537(n,k) - A277537(n,k-1) for k>0, T(n,0) = A000007(n).
%F A277536 T(n,k) = n * A295027(n,k) for n,k > 0.
%e A277536 Triangle T(n,k) begins:
%e A277536   1;
%e A277536   0, 1;
%e A277536   0, 0,   2;
%e A277536   0, 0,   3,     6;
%e A277536   0, 0,   8,    24,     24;
%e A277536   0, 0,  10,   170,    180,    120;
%e A277536   0, 0,  54,   900,   1980,   1440,    720;
%e A277536   0, 0, -42,  6566,  19530,  21840,  12600,   5040;
%e A277536   0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320;
%e A277536   ...
%p A277536 f:= proc(n) option remember; `if`(n<0, 0,
%p A277536       `if`(n=0, 1, (x+1)^f(n-1)))
%p A277536     end:
%p A277536 T:= (n, k)-> n!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
%p A277536 seq(seq(T(n, k), k=0..n), n=0..12);
%p A277536 # second Maple program:
%p A277536 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
%p A277536       -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
%p A277536       (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
%p A277536     end:
%p A277536 T:= (n, k)-> b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))):
%p A277536 seq(seq(T(n, k), k=0..n), n=0..12);
%t A277536 f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
%t A277536 T[n_, k_] := n!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
%t A277536 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
%t A277536 (* second program: *)
%t A277536 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 A277536 T[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k - 1, n]]]);
%t A277536 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A277536 Columns k=0-2 give: A000007, A063524, A005727 (for n>1).
%Y A277536 Main diagonal gives A000142.
%Y A277536 Row sums give A033917.
%Y A277536 T(n+1,n)/3 gives A005990.
%Y A277536 T(2n,n) gives A290023.
%Y A277536 Cf. A277537, A295027.
%K A277536 sign,tabl
%O A277536 0,6
%A A277536 _Alois P. Heinz_, Oct 19 2016