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%I M3960 N1634 #77 Jan 15 2023 02:43:07
%S 1,5,30,210,1680,15120,151200,1663200,19958400,259459200,3632428800,
%T 54486432000,871782912000,14820309504000,266765571072000,
%U 5068545850368000,101370917007360000,2128789257154560000,46833363657400320000,1077167364120207360000
%N a(n) = n!/24.
%C The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=5) ~ exp(-x)/x*(1 - 5/x + 30/x^2 - 210/x^3 + 1680/x^4 - 15120/x^5 + 151200/x^6 - 1663200/x^7 + ...) leads to this sequence. 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="/A001720/b001720.txt">Table of n, a(n) for n = 4..300</a>
%H S. Cerdá, <a href="http://arxiv.org/abs/1504.06694">A simple scheme to find the solutions to Brocard-Ramanujan Diophantine Equation n! + 1 = m^2</a>, arXiv:1504.06694 [math.NT], 2015.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=264">Encyclopedia of Combinatorial Structures 264</a>.
%H Wolfdieter Lang, <a href="http://www.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/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.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 a(n)= A049353(n-3, 1) (first column of triangle).
%F E.g.f. if offset 0: 1/(1-x)^5.
%F a(n) = A173333(n,4). - _Reinhard Zumkeller_, Feb 19 2010
%F a(n) = A245334(n,n-4) / 5. - _Reinhard Zumkeller_, Aug 31 2014
%F G(x) = (1 - (1 + x)^(-4)) / 4 = x - 5 x^2/2! + 30 x^3/3! - ..., an e.g.f. for this signed sequence (for n!/4!), is the compositional inverse of H(x) = (1 - 4*x)^(-1/4) - 1 = x + 5 x^2/2! + 45 x^3/3! + ..., an e.g.f. for A007696. Cf. A094638, A001710 (for n!/2!), and A001715 (for n!/3!). Cf. columns of A094587, A173333, and A213936 and rows of A138533. - _Tom Copeland_, Dec 27 2019
%F E.g.f.: x^4 / (4! * (1 - x)). - _Ilya Gutkovskiy_, Jul 09 2021
%F From _Amiram Eldar_, Jan 15 2023: (Start)
%F Sum_{n>=4} 1/a(n) = 24*e - 64.
%F Sum_{n>=4} (-1)^n/a(n) = 24/e - 8. (End)
%t a[n_]:=n!/24; (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%t Range[4,30]!/24 (* _Harvey P. Dale_, Jul 27 2021 *)
%o (Magma) [Factorial(n)/24: n in [4..25]]; // _Vincenzo Librandi_, Jul 20 2011
%o (PARI) a(n)=n!/24 \\ _Charles R Greathouse IV_, Jan 12 2012
%o (Haskell)
%o a001720 = (flip div 24) . a000142 -- _Reinhard Zumkeller_, Aug 31 2014
%Y Cf. A000142, A049353, A049459, A051338, A245334.
%Y Cf. A001710, A001715, A007696, A094638.
%Y Cf. A094587, A130534, A138533, A163931, A173333, A213936.
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
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