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A298605 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. 1
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, 910, 105, 7, 0, 944, 22036, 37352, 12740, 2072, 168, 8, 0, -5112, 182628, 441756, 200781, 37800, 4158, 252, 9, 0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Eric Weisstein's World of Mathematics, Power Tower

Wikipedia, Knuth's up-arrow notation

Wikipedia, Tetration

FORMULA

T(n,k) = n!/(k-1)! * [x^n] ((x+1)^^k - (x+1)^^(k-1)).

T(n,k) = 1/(k-1)! * [(d/dx)^n (x^^k - x^^(k-1))]_{x=1}.

T(n,k) = 1/(k-1)! * A277536(n,k).

T(n,k) = n/(k-1)! * A295027(n,k).

EXAMPLE

Triangle T(n,k) begins:

  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,     910,    105,     7;

  0,   944,   22036,   37352,   12740,   2072,   168,    8;

  0, -5112,  182628,  441756,  200781,  37800,  4158,  252,   9;

  0, 47160, 1488240, 5765540, 3282300, 747390, 94500, 7620, 360, 10;

MAPLE

f:= proc(n) option remember; `if`(n<0, 0,

      `if`(n=0, 1, (x+1)^f(n-1)))

    end:

T:= (n, k)-> n!/(k-1)!*coeff(series(f(k)-f(k-1), x, n+1), x, n):

seq(seq(T(n, k), k=1..n), n=1..10);

# 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))))/(k-1)!:

seq(seq(T(n, k), k=1..n), n=1..10);

MATHEMATICA

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];

T[n_, k_] := n!/(k - 1)!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];

Table[T[n, k], {n, 1, 10}, { k, 1, 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]]])/(k-1)!;

Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 03 2018, from Maple *)

CROSSREFS

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

Main diagonal gives A000027.

Cf. A277536, A295027.

Sequence in context: A154344 A134409 A327878 * A180013 A094067 A094112

Adjacent sequences:  A298602 A298603 A298604 * A298606 A298607 A298608

KEYWORD

sign,tabl

AUTHOR

Alois P. Heinz, Jan 22 2018

STATUS

approved

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Last modified February 20 07:54 EST 2020. Contains 332069 sequences. (Running on oeis4.)