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%I M4243 N1772 #59 Jan 15 2023 02:43:25
%S 1,6,42,336,3024,30240,332640,3991680,51891840,726485760,10897286400,
%T 174356582400,2964061900800,53353114214400,1013709170073600,
%U 20274183401472000,425757851430912000,9366672731480064000,215433472824041472000,5170403347776995328000
%N a(n) = n!/5!.
%C The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - _Johannes W. Meijer_, Oct 20 2009
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A001725/b001725.txt">Table of n, a(n) for n = 5..300</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=265">Encyclopedia of Combinatorial Structures 265</a>.
%H Wolfdieter Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/54/107.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling. II</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F E.g.f. if offset 0: 1/(1-x)^6.
%F a(n) = A173333(n,5). - _Reinhard Zumkeller_, Feb 19 2010
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013
%F G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 21 2013
%F a(n) = A245334(n,n-5) / 6. - _Reinhard Zumkeller_, Aug 31 2014
%F E.g.f.: x^5 / (5! * (1 - x)). - _Ilya Gutkovskiy_, Jul 09 2021
%F From _Amiram Eldar_, Jan 15 2023: (Start)
%F Sum_{n>=5} 1/a(n) = 120*e - 325.
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)
%t lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 25 2008 *)
%t Range[5,30]!/120 (* _Harvey P. Dale_, Dec 20 2014 *)
%o (PARI) a(n)=n!/120 \\ _Charles R Greathouse IV_, Jul 19 2011
%o (Magma) [Factorial(n)/120: n in [5..25]]; // _Vincenzo Librandi_, Jul 20 2011
%o (Haskell)
%o a001725 = (flip div 120) . a000142 -- _Reinhard Zumkeller_, Aug 31 2014
%Y a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).
%Y Cf. A245334, A000142.
%K nonn,easy
%O 5,2
%A _N. J. A. Sloane_
%E More terms from _Harvey P. Dale_, Dec 20 2014
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