This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356009 #10 Aug 11 2025 05:32:05
%S A356009 1,4,15,73,390,2641,19208,164585,1541746,16158341,181370552,
%T A356009 2283224065,30160914446,434715492485,6655132252876,109315669969217,
%U A356009 1879289179364690,34719396682318021,666070910669770400,13590051478686198401,289043813095242038422
%N A356009 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * (k/d)!).
%F A356009 E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1)/k.
%F A356009 E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1-x^k)/k!.
%t A356009 Table[n! * Sum[Sum[1/(d*(k/d)!), {d,Divisors[k]}], {k,1,n}], {n,1,25}] (* _Vaclav Kotesovec_, Aug 11 2025 *)
%o A356009 (PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!)));
%o A356009 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp(x^k)-1)/k)/(1-x)))
%o A356009 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k)/k!)/(1-x)))
%Y A356009 Cf. A087906, A356004.
%K A356009 nonn
%O A356009 1,2
%A A356009 _Seiichi Manyama_, Jul 23 2022