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A000399
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Unsigned Stirling numbers of first kind s(n,3).
(Formerly M4218 N1762)
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18
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1, 6, 35, 225, 1624, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| Number of permutations of n elements with exactly 3 cycles.
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=1) ~ exp(-x)/x^3*(1 - 6/x + 35/x^2 - 225/x^3 + 1624/x^4 - 13132/x^5 + ...) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
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).
Shanzhen Gao, Permutations with Restricted Structure (in preparation). - Shanzhen Gao (shanzhengao(AT)yahoo.com), Sep 14 2010
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LINKS
| T. D. Noe, Table of n, a(n) for n=3..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 32
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FORMULA
| Let P(n+1,X)=(X+1)(X+2)(X+3)...(X+n+1); then a(n) is the coefficient of X^2; or a(n)=P''(n+1,0)/2! - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2002
E.g.f. -log(1-x)^3/3!
a(n) is coefficient of x^(n+3) in (-log(1-x))^3, multiplied by (n+3)!/6.
[(sum(1/i, i=1..n-1)^2-sum(1/i^2, i=1..n-1)]*(n-1)!/2. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000
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EXAMPLE
| (-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...
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MATHEMATICA
| a=Log[1/(1-x)]; Range[0, 20]! CoefficientList[Series[a^3/3!, {x, 0, 20}], x]
f[n_] := Abs@ StirlingS1[n, 3]; Array[f, 19, 3]
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PROG
| (MuPAD) f := proc(n) option remember; begin n^3*f(n-3)-(3*n^2+3*n+1)*f(n-2)+3*(n+1)*f(n-1) end_proc: f(0) := 1: f(1) := 6: f(2) := 35:
(PARI) for(n=2, 50, print1(polcoeff(prod(i=1, n, x+i), 2, x), ", "))
sage: [stirling_number1(i+2, 3) for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
(MAGMA) A000399:=func< n | Abs(StirlingFirst(n, 3)) >; [ A000399(n): n in [3..25] ]; - Klaus Brockhaus, Jan 14 2011
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CROSSREFS
| Cf. A000254, A000454, A000482, A001233, A008275 (Stirling1 triangle).
Sequence in context: A180033 A144638 A117671 * A081051 A145145 A187443
Adjacent sequences: A000396 A000397 A000398 * A000400 A000401 A000402
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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