A001725 a(n) = n!/5!.

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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 = 5..300
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.
• 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. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 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

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).

Cf. A245334, A000142.

Programs

Haskell

a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

Magma

[Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5,30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

PARI

a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011

Formula

E.g.f. if offset 0: 1/(1-x)^6.

a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010

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

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

a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014

E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=5} 1/a(n) = 120*e - 325.

Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

Extensions

More terms from Harvey P. Dale, Dec 20 2014