A290023 a(n) is the 2n-th derivative of the difference between the n-th tetration of x (power tower of order n) and its predecessor (or 0 if n=0) at x=1.

1, 0, 8, 900, 224112, 78775200, 40518181440, 28340179227360, 26078095792869120, 30544708065077606400, 44428404658605222528000, 78604530683773395984883200, 166295474965751756924207462400, 414658685362517268992110471680000, 1203746810444949373635048911870976000

Offset: 0

Author

Alois P. Heinz, Nov 10 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Knuth's up-arrow notation
• Wikipedia, Tetration

Crossrefs

Cf. A277536.

Programs

Maple

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
a:= n-> (2*n)!*coeff(series(f(n)-f(n-1), x, 2*n+1), x, 2*n):
seq(a(n), n=0..15);
# 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:
a:= n-> b(2*n, n) -`if`(n=0, 0, b(2*n, n-1)):
seq(a(n), n=0..15);

Mathematica

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
a[n_] := (2*n)!*SeriesCoefficient[f[n] - f[n - 1], {x, 0, 2*n}];
Table[a[n], {n, 0, 15}]
(* 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}]]];
a[n_] := b[2*n, n] - If[n == 0, 0, b[2*n, n - 1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = (2n)! * [x^(2n)] (x+1)^^n - (x+1)^^(n-1) for n>0, a(0) = 1.

a(n) = [(d/dx)^(2n) (x^^n - x^^(n-1))]_{x=1} for n>0, a(0) = 1.

a(n) = A277536(2n,n).