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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.

%I A290023 #28 Feb 16 2025 08:33:49
%S A290023 1,0,8,900,224112,78775200,40518181440,28340179227360,
%T A290023 26078095792869120,30544708065077606400,44428404658605222528000,
%U A290023 78604530683773395984883200,166295474965751756924207462400,414658685362517268992110471680000,1203746810444949373635048911870976000
%N 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.
%H A290023 Alois P. Heinz, <a href="/A290023/b290023.txt">Table of n, a(n) for n = 0..100</a>
%H A290023 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A290023 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A290023 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A290023 a(n) = (2n)! * [x^(2n)] (x+1)^^n - (x+1)^^(n-1) for n>0, a(0) = 1.
%F A290023 a(n) = [(d/dx)^(2n) (x^^n - x^^(n-1))]_{x=1} for n>0, a(0) = 1.
%F A290023 a(n) = A277536(2n,n).
%p A290023 f:= proc(n) option remember; `if`(n<0, 0,
%p A290023       `if`(n=0, 1, (x+1)^f(n-1)))
%p A290023     end:
%p A290023 a:= n-> (2*n)!*coeff(series(f(n)-f(n-1), x, 2*n+1), x, 2*n):
%p A290023 seq(a(n), n=0..15);
%p A290023 # second Maple program:
%p A290023 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
%p A290023       -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
%p A290023       (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
%p A290023     end:
%p A290023 a:= n-> b(2*n, n) -`if`(n=0, 0, b(2*n, n-1)):
%p A290023 seq(a(n), n=0..15);
%t A290023 f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
%t A290023 a[n_] := (2*n)!*SeriesCoefficient[f[n] - f[n - 1], {x, 0, 2*n}];
%t A290023 Table[a[n], {n, 0, 15}]
%t A290023 (* Second program: *)
%t A290023 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 A290023 a[n_] := b[2*n, n] - If[n == 0, 0, b[2*n, n - 1]];
%t A290023 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A290023 Cf. A277536.
%K A290023 nonn
%O A290023 0,3
%A A290023 _Alois P. Heinz_, Nov 10 2017