A001720 a(n) = n!/24.

1, 5, 30, 210, 1680, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 871782912000, 14820309504000, 266765571072000, 5068545850368000, 101370917007360000, 2128789257154560000, 46833363657400320000, 1077167364120207360000

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 4..300
• S. Cerdá, A simple scheme to find the solutions to Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, arXiv:1504.06694 [math.NT], 2015.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 264.
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
• D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
• Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
• Index entries for sequences related to factorial numbers.
• Index to divisibility sequences.

Crossrefs

Cf. A000142, A049353, A049459, A051338, A245334.

Cf. A001710, A001715, A007696, A094638.

Cf. A094587, A130534, A138533, A163931, A173333, A213936.

Programs

Haskell

a001720 = (flip div 24) . a000142 -- Reinhard Zumkeller, Aug 31 2014

Magma

[Factorial(n)/24: n in [4..25]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

a[n_]:=n!/24; (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[4,30]!/24 (* Harvey P. Dale, Jul 27 2021 *)

PARI

a(n)=n!/24 \\ Charles R Greathouse IV, Jan 12 2012

Formula

a(n)= A049353(n-3, 1) (first column of triangle).

E.g.f. if offset 0: 1/(1-x)^5.

a(n) = A173333(n,4). - Reinhard Zumkeller, Feb 19 2010

a(n) = A245334(n,n-4) / 5. - Reinhard Zumkeller, Aug 31 2014

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

E.g.f.: x^4 / (4! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=4} 1/a(n) = 24*e - 64.

Sum_{n>=4} (-1)^n/a(n) = 24/e - 8. (End)