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A277538 n-th derivative of the seventh tetration of x (power tower of order 7) x^^7 at x=1.

%I A277538 #17 Feb 16 2025 08:33:37
%S A277538 1,1,2,9,56,480,5094,65534,944488,15359184,274118880,5369469072,
%T A277538 114055774392,2618129199504,64505645760360,1699067695202040,
%U A277538 47625773113856064,1415693345589370368,44474719907550007872,1472352462644660486208,51227002322058534554880
%N A277538 n-th derivative of the seventh tetration of x (power tower of order 7) x^^7 at x=1.
%H A277538 Alois P. Heinz, <a href="/A277538/b277538.txt">Table of n, a(n) for n = 0..400</a>
%H A277538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A277538 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A277538 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A277538 E.g.f.: (x+1)^^7.
%p A277538 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A277538 a:= n-> n!*coeff(series(f(7), x, n+1), x, n):
%p A277538 seq(a(n), n=0..25);
%t A277538 f[0] = 1; f[n_] := f[n] = (x + 1)^f[n - 1];
%t A277538 a[n_] := n! SeriesCoefficient[f[7], {x, 0, n}];
%t A277538 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y A277538 Column k=7 of A277537.
%Y A277538 Cf. A215703, A295107.
%K A277538 nonn
%O A277538 0,3
%A A277538 _Alois P. Heinz_, Oct 19 2016