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A295110 a(n) = (1/n) times the n-th derivative of the tenth tetration of x (power tower of order 10) x^^10 at x=1.

%I A295110 #10 Feb 16 2025 08:33:51
%S A295110 1,1,3,14,96,849,9362,123101,1888016,33066768,648254352,14047013386,
%T A295110 332435753328,8525497523100,235255431749816,6948357529667124,
%U A295110 218632693287458304,7300788870920480864,257844842264845841472,9602530635731056455744,376083658114082740084224
%N A295110 a(n) = (1/n) times the n-th derivative of the tenth tetration of x (power tower of order 10) x^^10 at x=1.
%H A295110 Alois P. Heinz, <a href="/A295110/b295110.txt">Table of n, a(n) for n = 1..407</a>
%H A295110 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A295110 Wikipedia, <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation">Knuth's up-arrow notation</a>
%H A295110 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F A295110 a(n) = 1/n * [(d/dx)^n x^^10]_{x=1}.
%F A295110 a(n) = (n-1)! * [x^n] (x+1)^^10.
%F A295110 a(n) = 1/n * A277541(n).
%p A295110 f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
%p A295110 a:= n-> (n-1)!*coeff(series(f(10), x, n+1), x, n):
%p A295110 seq(a(n), n=1..23);
%t A295110 f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
%t A295110 a[n_] := (n - 1)!*SeriesCoefficient[f[10], {x, 0, n}];
%t A295110 Array[a, 23] (* _Jean-François Alcover_, May 31 2018, from Maple *)
%Y A295110 Column k=10 of A295028.
%Y A295110 Cf. A277541.
%K A295110 nonn
%O A295110 1,3
%A A295110 _Alois P. Heinz_, Nov 14 2017