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 A259472 #58 Jan 08 2018 01:36:26
%S A259472 1,-2,-1,-4,-19,-110,-745,-5752,-49775,-476994,-5016069,-57462828,
%T A259472 -712732987,-9521244982,-136356161873,-2084860795232,-33907076207495,
%U A259472 -584602069590058,-10652917092110429,-204604743619641620,-4131502481607654739,-87507494737954740126
%N A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials.
%H A259472 Vaclav Kotesovec, <a href="/A259472/b259472.txt">Table of n, a(n) for n = 0..446</a>
%H A259472 L. Comtet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5619085m/f9.image">Sur les coefficients de l'inverse de la série formelle Sum n! t^n</a>, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
%H A259472 L. Comtet, <a href="/A003319/a003319_1.pdf">Series inversions</a>, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
%H A259472 R. K. Guy, <a href="/A003320/a003320.pdf">Letter to N. J. A. Sloane, Mar 1974</a>
%F A259472 From _Vaclav Kotesovec_, Aug 12 2015: (Start)
%F A259472 G.f.: (1/Sum(k! x^k))^2.
%F A259472 Expansion of (1-g(x))^2, where g(x) is the g.f. of A003319.
%F A259472 a(n) ~ -2*n! * (1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - 31291/n^7 - 395360/n^8 - 5544754/n^9 - 85427259/n^10), for coefficients see A261214.
%F A259472 For n>0, a(n) = A059439(n) - 2*A003319(n).
%F A259472 For n>0, a(n) = Sum_{k=1..n} A260503(k) * Stirling1(n-1, k-1).
%F A259472 (End)
%e A259472 A003319(n) / n! ~ 1 - 2/n - 1/(n*(n-1)) - 4/(n*(n-1)*(n-2)) - 19/(n*(n-1)*(n-2)*(n-3)) - 110/(n*(n-1)*(n-2)*(n-3)*(n-4)) - 745/(n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)) - ... [coefficients are A259472]
%e A259472 A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ... [coefficients are A260503]
%t A259472 CoefficientList[Series[1/Sum[k! * x^k, {k, 0, 20}]^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 03 2015 *)
%t A259472 CoefficientList[Assuming[Element[x,Reals], Series[E^(2/x) * x^2 / ExpIntegralEi[1/x]^2, {x,0,25}]], x] (* _Vaclav Kotesovec_, Aug 03 2015 *)
%Y A259472 Cf. A003319, A260503, A261214, A261239, A261253, A261254, A059439.
%K A259472 sign
%O A259472 0,2
%A A259472 _N. J. A. Sloane_, Jul 03 2015, following a suggestion from _R. K. Guy_, Apr 29 1974
%E A259472 More terms from _Vaclav Kotesovec_, Aug 01 2015
%E A259472 New name from _Vaclav Kotesovec_, Aug 12 2015
%E A259472 Entry revised by _Vaclav Kotesovec_, Aug 12 2015