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.

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, 1440, 720, 0, 0, -42, 6566, 19530, 21840, 12600, 5040, 0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320, 0, 0, -5112, 365256, 2650536, 4818744, 4536000, 2993760, 1270080, 362880

Offset

0,6

Comments

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.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Eric Weisstein's World of Mathematics, Power Tower

Wikipedia, Knuth's up-arrow notation

Wikipedia, Tetration

Formula

E.g.f. of column k>0: (x+1)^^k - (x+1)^^(k-1), e.g.f. of column k=0: 1.

T(n,k) = [(d/dx)^n (x^^k - x^^(k-1))]_{x=1} for k>0, T(n,0) = A000007(n).

T(n,k) = A277537(n,k) - A277537(n,k-1) for k>0, T(n,0) = A000007(n).

T(n,k) = n * A295027(n,k) for n,k > 0.

Example

Triangle T(n,k) begins:
  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,   1440,    720;
  0, 0, -42,  6566,  19530,  21840,  12600,   5040;
  0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320;
  ...

Maple

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
T:= (n, k)-> n!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..12);
# 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:
T:= (n, k)-> b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := n!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
(* second program: *)
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[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k - 1, n]]]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2018, from Maple *)

Crossrefs

Columns k=0-2 give: A000007, A063524, A005727 (for n>1).

Main diagonal gives A000142.

Row sums give A033917.

T(n+1,n)/3 gives A005990.

T(2n,n) gives A290023.

Cf. A277537, A295027.

Sequence in context: A336309 A336255 A244120 * A348849 A351142 A269159

Adjacent sequences: A277533 A277534 A277535 * A277537 A277538 A277539

Keyword

sign,tabl

Author

Alois P. Heinz, Oct 19 2016

Status

approved