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%I M4436 N1876 #48 Jan 15 2023 02:42:09
%S 1,7,56,504,5040,55440,665280,8648640,121080960,1816214400,
%T 29059430400,494010316800,8892185702400,168951528345600,
%U 3379030566912000,70959641905152000,1561112121913344000,35905578804006912000,861733891296165888000,21543347282404147200000
%N a(n) = n!/6!.
%C The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=7) ~ exp(-x)/x*(1 - 7/x + 56/x^2 - 504/x^3 + 5040/x^4 - 55440/x^5 + 665280/x^6 - 8648640/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="/A001730/b001730.txt">Table of n, a(n) for n = 6..300</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=266">Encyclopedia of Combinatorial Structures 266</a>.
%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 <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 a(n)= A051339(n-6, 0)*(-1)^n (first unsigned column of triangle).
%F E.g.f.: x^6/(6!*(1-x)). [corrected by _Alois P. Heinz_, Jul 09 2021]
%F a(n) = A173333(n,6). - _Reinhard Zumkeller_, Feb 19 2010
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+7)/(x*(k+7) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013
%F a(n) = A245334(n,n-6) / 7. - _Reinhard Zumkeller_, Aug 31 2014
%F From _Amiram Eldar_, Jan 15 2023: (Start)
%F Sum_{n>=6} 1/a(n) = 720*e - 1956.
%F Sum_{n>=6} (-1)^n/a(n) = 720/e - 264. (End)
%t a[n_]:=n!/6!; Array[a,4!,6] (* _Vladimir Joseph Stephan Orlovsky_, Oct 25 2009 *)
%o (Magma) [Factorial(n)/720: n in [6..25]]; // _Vincenzo Librandi_, Jul 20 2011
%o (PARI) a(n)=n!/720 \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Haskell)
%o a001730 = (flip div 720) . a000142 -- _Reinhard Zumkeller_, Aug 31 2014
%Y Cf. A051338, A051339, A051379.
%Y Cf. A130534, A163931, A173333, A245334, A000142.
%K nonn,easy
%O 6,2
%A _N. J. A. Sloane_
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