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A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.

%I A355886 #24 Aug 12 2025 02:41:08
%S A355886 1,5,22,125,746,5677,44780,420401,4206970,47543141,562891352,
%T A355886 7573655905,104684547566,1596368400005,25482043382476,439969180782017,
%U A355886 7835163501390290,151712475696833221,3004182138648663200,63854641556089628801,1400563708969910620822
%N A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.
%H A355886 Seiichi Manyama, <a href="/A355886/b355886.txt">Table of n, a(n) for n = 1..448</a>
%H A355886 Vaclav Kotesovec, <a href="/A355886/a355886.jpg">Plot of a(n)/(n!*n) for n = 1..10000</a>
%F A355886 E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k! * (1 - x^k)).
%F A355886 E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1).
%F A355886 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/d! = n! * Sum_{k=1..n} A057625(k)/k!. - _Seiichi Manyama_, Aug 08 2022
%F A355886 a(n) ~ A229837 * n! * n. - _Vaclav Kotesovec_, Aug 11 2025
%t A355886 Table[n! * Sum[Floor[n/k]/k!, {k,1,n}], {n,1,25}] (* _Vaclav Kotesovec_, Aug 11 2025 *)
%o A355886 (PARI) a(n) = n!*sum(k=1, n, n\k/k!);
%o A355886 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k)))/(1-x)))
%o A355886 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(x^k)-1)/(1-x)))
%o A355886 (PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ _Seiichi Manyama_, Aug 08 2022
%Y A355886 Cf. A006218, A057625, A356010, A356458, A229837.
%Y A355886 Cf. A356009, A356459.
%K A355886 nonn
%O A355886 1,2
%A A355886 _Seiichi Manyama_, Jul 20 2022