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A001715
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n!/6.
(Formerly M3566 N1445)
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36
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1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| 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)
a(n) = A173333(n,3). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 19 2010]
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REFERENCES
| 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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 3..200
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 to divisibility sequences
Index entries for sequences related to factorial numbers
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FORMULA
| E.g.f. if offset 0: 1/(1-x)^4.
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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 (zerinvarylajos(AT)yahoo.com), Jun 19 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]
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MATHEMATICA
| a[n_]:=n!/6; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]
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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
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CROSSREFS
| a(n) = A049352(n-2, 1) (first column of triangle). Cf. A049458, A049460.
Cf. A034472, A130128.
Sequence in context: A093123 A092055 A187848 * A020028 A020118 A009351
Adjacent sequences: A001712 A001713 A001714 * A001716 A001717 A001718
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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