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A001715 a(n) = n!/6.

%I A001715 M3566 N1445 #90 Jan 15 2023 02:42:45
%S A001715 1,4,20,120,840,6720,60480,604800,6652800,79833600,1037836800,
%T A001715 14529715200,217945728000,3487131648000,59281238016000,
%U A001715 1067062284288000,20274183401472000,405483668029440000,8515157028618240000,187333454629601280000,4308669456480829440000
%N A001715 a(n) = n!/6.
%C A001715 The numbers (4, 20, 120, 840, 6720, ...) arise from the divisor values in the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ... *(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) (which covers the following sequences: A000578, A000537, A024166, A101094, A101097, A101102). - _Alexander R. Povolotsky_, May 17 2008
%C A001715 a(n) is also the number of decreasing 3-cycles in the decomposition of permutations as product of disjoint cycles, a(3)=1, a(4)=4, a(5)=20. - _Wenjin Woan_, Dec 21 2008
%C A001715 Equals eigensequence of triangle A130128 reflected. - _Gary W. Adamson_, Dec 23 2008
%C A001715 a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - _Geoffrey Critzer_, Apr 26 2009
%C A001715 From _Johannes W. Meijer_, Oct 20 2009: (Start)
%C A001715 The asymptotic expansion of the higher order exponential integral E(x,m=1,n=4) ~ exp(-x)/x*(1 - 4/x + 20/x^2 - 120/x^3 + 840/x^4 - 6720/x^5 + 60480/x^6 - 604800/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.
%C A001715 (End)
%D A001715 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001715 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001715 Vincenzo Librandi, <a href="/A001715/b001715.txt">Table of n, a(n) for n = 3..200</a>
%H A001715 Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, <a href="https://arxiv.org/abs/1812.02955">Mixed restricted Stirling numbers</a>, arXiv:1812.02955 [math.CO], 2018.
%H A001715 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=263">Encyclopedia of Combinatorial Structures 263</a>.
%H A001715 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 A001715 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, 77 pp.
%H A001715 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 A001715 A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%H A001715 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%H A001715 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A001715 a(n) = A049352(n-2, 1) (first column of triangle).
%F A001715 E.g.f. if offset 0: 1/(1-x)^4.
%F A001715 a(n) = A173333(n,3). - _Reinhard Zumkeller_, Feb 19 2010
%F A001715 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+4)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013
%F A001715 G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Aug 26 2013
%F A001715 a(n) = A245334(n,n-3) / 4. - _Reinhard Zumkeller_, Aug 31 2014
%F A001715 From _Peter Bala_, May 22 2017: (Start)
%F A001715 The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (4*x - 1)*A(x) + 1 = 0.
%F A001715 G.f. as an S-fraction: A(x) = 1/(1 - 4*x/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - ... - (n + 3)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
%F A001715 A(x) = 1/(1 - 3*x - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - 6*x/(1 - ... - n*x/(1 - (n+3)*x/(1 - ... ))))))))). (End)
%F A001715 H(x) = (1 - (1 + x)^(-3)) / 3 = x - 4 x^2/2! + 20 x^3/3! - ... is an e.g.f. of the signed sequence (n!/4!), which is the compositional inverse of G(x) = (1 - 3*x)^(-1/3) - 1, an e.g.f. for A007559. Cf. A094638, A001710 (for n!/2!), and A001720 (for n!/4!). Cf. columns of A094587, A173333, and A213936 and rows of A138533.- _Tom Copeland_, Dec 27 2019
%F A001715 E.g.f.: x^3 / (3! * (1 - x)). - _Ilya Gutkovskiy_, Jul 09 2021
%F A001715 From _Amiram Eldar_, Jan 15 2023: (Start)
%F A001715 Sum_{n>=3} 1/a(n) = 6*e - 15.
%F A001715 Sum_{n>=3} (-1)^(n+1)/a(n) = 3 - 6/e. (End)
%p A001715 f := proc(n) n!/6; end;
%p A001715 BB:= [S, {S = Prod(Z,Z,C), C = Union(B,Z,Z), B = Prod(Z,C)}, labelled]: seq(combstruct[count](BB, size=n)/12, n=3..20); # _Zerinvary Lajos_, Jun 19 2008
%p A001715 G(x):=1/(1-x)^4: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..16); # _Zerinvary Lajos_, Apr 01 2009
%t A001715 a[n_]:=n!/6; (*_Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%t A001715 Range[3,30]!/6 (* _Harvey P. Dale_, Aug 12 2012 *)
%o A001715 (Magma) [Factorial(n)/6: n in [3..30]]; // _Vincenzo Librandi_, Jun 20 2011
%o A001715 (PARI) a(n)=n!/6 \\ _Charles R Greathouse IV_, Jan 12 2012
%o A001715 (Haskell)
%o A001715 a001715 = (flip div 6) . a000142 -- _Reinhard Zumkeller_, Aug 31 2014
%Y A001715 Cf. A049352, A049458, A049460.
%Y A001715 Cf. A034472, A130128.
%Y A001715 Cf. A245334, A000142, A111530.
%Y A001715 Cf. A001710, A001720, A007759, A094638.
%Y A001715 Cf. A094587, A138533, A173333, A213936.
%K A001715 nonn,easy
%O A001715 3,2
%A A001715 _N. J. A. Sloane_
%E A001715 More terms from _Harvey P. Dale_, Aug 12 2012