3,2

Those 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 following sequences: A000578, A000537, A024166, A101094, A101097, A101102) - Alexander R. Povolotsky (pevnev(AT)juno.com), 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. [From Wenjin Woan (wjwoan(AT)hotmail.com), Dec 21 2008]

Equals eigensequence of triangle A130128 reflected. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008]

a(n) is the number of n-permutations having 1,2,and 3 in three distinct cycles. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 26 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), 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)

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

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).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 263

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

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

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

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

seq(mul((k), k=4..n), n=3..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]*(n+2)^2) od: seq(sqrt(a[n]), n=1..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 04 2008

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 (zerinvarylajos(AT)yahoo.com), Jun 19 2008

a:=n->mul(denom((k+1)/(k+2) ), k=2..n): seq(a(n), n=1..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 2008

a:=n->mul(numer((k+1)/(k+2) ), k=3..n): seq(a(n), n=2..18); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 2008

restart: 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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 01 2009]

a[n_]:=n!/6; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]

a(n) = A049352(n-2, 1) (first column of triangle). Cf. A049458, A049460.

Cf. A034472.

A130128 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008]

Sequence in context: A046729 A093123 A092055 this_sequence A020028 A020118 A009351

Adjacent sequences: A001712 A001713 A001714 this_sequence A001716 A001717 A001718

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).