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a(n) = n! * Sum_{k=1..n} 1/(k! * floor(n/k)!).
7

%I #21 Jul 23 2022 09:53:27

%S 1,2,5,12,57,158,1101,5442,28811,212502,2337513,9422306,122489967,

%T 1654319046,13917499277,111631450818,1897734663891,23705612782022,

%U 450406642858401,3091477152208002,51404897928720023,1130752882197523686,26007316290543044757

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

%H Seiichi Manyama, <a href="/A355991/b355991.txt">Table of n, a(n) for n = 1..464</a>

%F E.g.f.: (1/(1-x)) * Sum_{k>0} (1 - x^k) * (exp(x^k) - 1)/k!.

%t a[n_] := n! * Sum[1/(k! * Floor[n/k]!), {k, 1, n}]; Array[a, 23] (* _Amiram Eldar_, Jul 22 2022 *)

%o (PARI) a(n) = n!*sum(k=1, n, 1/(k!*(n\k)!));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (1-x^k)*(exp(x^k)-1)/k!)/(1-x)))

%Y Row sums of A355996.

%Y Cf. A121860, A355886, A355987, A355989, A355990.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 22 2022