A001715 a(n) = n!/6. (Formerly M3566 N1445)

1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000, 405483668029440000, 8515157028618240000, 187333454629601280000, 4308669456480829440000

Offset

3,2

Comments

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

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

Equals eigensequence of triangle A130128 reflected. - Gary W. Adamson, Dec 23 2008

a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - Geoffrey Critzer, Apr 26 2009

From Johannes W. Meijer, Oct 20 2009: (Start)

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.

(End)

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 = 3..200

Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 263.

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, 77 pp.

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.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Index entries for sequences related to factorial numbers.

Index to divisibility sequences.

Formula

a(n) = A049352(n-2, 1) (first column of triangle).

E.g.f. if offset 0: 1/(1-x)^4.

a(n) = A173333(n,3). - Reinhard Zumkeller, Feb 19 2010

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

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

a(n) = A245334(n,n-3) / 4. - Reinhard Zumkeller, Aug 31 2014

From Peter Bala, May 22 2017: (Start)

The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (4*x - 1)*A(x) + 1 = 0.

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).

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)

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

E.g.f.: x^3 / (3! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021

From Amiram Eldar, Jan 15 2023: (Start)

Sum_{n>=3} 1/a(n) = 6*e - 15.

Sum_{n>=3} (-1)^(n+1)/a(n) = 3 - 6/e. (End)

Maple

f := proc(n) n!/6; end;
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
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

Mathematica

a[n_]:=n!/6; (*Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[3, 30]!/6 (* Harvey P. Dale, Aug 12 2012 *)

Prog

(Magma) [Factorial(n)/6: n in [3..30]]; // Vincenzo Librandi, Jun 20 2011
(PARI) a(n)=n!/6 \\ Charles R Greathouse IV, Jan 12 2012
(Haskell)
a001715 = (flip div 6) . a000142 -- Reinhard Zumkeller, Aug 31 2014

Crossrefs

Cf. A049352, A049458, A049460.

Cf. A034472, A130128.

Cf. A245334, A000142, A111530.

Cf. A001710, A001720, A007759, A094638.

Cf. A094587, A138533, A173333, A213936.

Sequence in context: A092055 A187848 A370653 * A304069 A020028 A020118

Adjacent sequences: A001712 A001713 A001714 * A001716 A001717 A001718

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Aug 12 2012

Status

approved