A001730 a(n) = n!/6!.

1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960, 1816214400, 29059430400, 494010316800, 8892185702400, 168951528345600, 3379030566912000, 70959641905152000, 1561112121913344000, 35905578804006912000, 861733891296165888000, 21543347282404147200000

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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 = 6..300
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 266.
• 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.
• Index entries for sequences related to factorial numbers.
• Index to divisibility sequences.

Crossrefs

Cf. A051338, A051339, A051379.

Cf. A130534, A163931, A173333, A245334, A000142.

Programs

Haskell

a001730 = (flip div 720) . a000142 -- Reinhard Zumkeller, Aug 31 2014

Magma

[Factorial(n)/720: n in [6..25]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

a[n_]:=n!/6!; Array[a,4!,6] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2009 *)

PARI

a(n)=n!/720 \\ Charles R Greathouse IV, Jan 12 2012

Formula

a(n)= A051339(n-6, 0)*(-1)^n (first unsigned column of triangle).

E.g.f.: x^6/(6!*(1-x)). [corrected by Alois P. Heinz, Jul 09 2021]

a(n) = A173333(n,6). - Reinhard Zumkeller, Feb 19 2010

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

a(n) = A245334(n,n-6) / 7. - Reinhard Zumkeller, Aug 31 2014

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=6} 1/a(n) = 720*e - 1956.

Sum_{n>=6} (-1)^n/a(n) = 720/e - 264. (End)