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a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x).
3

%I #15 Feb 02 2018 07:00:39

%S 1,1,7,381,502789,33572762781,175123095782787181,

%T 99374457734129265819664221,8158897372191288496224413025490409437,

%U 124778468912108975502836576328262294089846582756189

%N a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x).

%H Seiichi Manyama, <a href="/A299036/b299036.txt">Table of n, a(n) for n = 0..30</a>

%F From _Vaclav Kotesovec_, Feb 02 2018: (Start)

%F a(n) ~ (n!)^n.

%F a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). (End)

%t Table[SeriesCoefficient[Product[1/(1 - k!*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Feb 02 2018 *)

%Y Cf. A000178, A036740.

%Y Cf. A007820, A298851, A299035.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 01 2018