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 A093344 #37 Feb 27 2022 22:56:59
%S A093344 0,1,4,17,84,485,3236,24609,210572,2004749,21033900,241237001,
%T A093344 3003349124,40345599957,581765196884,8963453118065,146969877361116,
%U A093344 2555361954692189,46963373856864092,909707559383702169,18524816853636447380,395634467245613474981
%N A093344 a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} 1/j!.
%H A093344 Alois P. Heinz, <a href="/A093344/b093344.txt">Table of n, a(n) for n = 0..400</a>
%F A093344 E.g.f.: exp(1)*(Ei(1,1-x)-Ei(1,1))/(1-x). - _Vladeta Jovovic_, May 05 2007
%F A093344 a(n) = Integral_{x=1..oo} exp(1-x)*(x^n*log(x) - n!/x) dx. - _Groux Roland_, Mar 12 2011
%F A093344 From _Vladimir Reshetnikov_, Oct 28 2015: (Start)
%F A093344 a(n) = exp(1)*(H(n)*n! + (Ei(-1)-gamma)*n! + hypergeom([n+1,n+1],[n+2,n+2],-1)/(n+1)^2), where H(n)*n! = A000254(n), -Ei(-1) is A099285, gamma is A001620.
%F A093344 Recurrence: a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3).
%F A093344 (End)
%F A093344 a(n) = n!*e*Sum_{k=1..n} Gamma(k,1)/k!. - _Robert Israel_, Oct 28 2015
%p A093344 f:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3)},a(n),remember):
%p A093344 seq(f(n),n=0..50); # _Robert Israel_, Oct 28 2015
%t A093344 Round@Table[E n! Sum[Gamma[k, 1]/k!, {k, 1, n}], {n, 0, 20}]
%t A093344 Round@Table[E ((HarmonicNumber[n] + ExpIntegralEi[-1] - EulerGamma) n! + HypergeometricPFQ[{n+1,n+1},{n+2,n+2},-1]/(n+1)^2), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%o A093344 (PARI) a(n) = n!*sum(i=1,n,1/i*sum(j=0,i-1,1/j!))
%Y A093344 Cf. A000254, A000774.
%Y A093344 Equals for n=>1 the row sums of A165674 and A093905. - _Johannes W. Meijer_, Oct 16 2009
%K A093344 nonn
%O A093344 0,3
%A A093344 _Ralf Stephan_, Apr 26 2004