login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000399 Unsigned Stirling numbers of first kind s(n,3).
(Formerly M4218 N1762)
21
1, 6, 35, 225, 1624, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

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, Oct 20 2009

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.

Shanzhen Gao, Permutations with Restricted Structure (in preparation). - Shanzhen Gao, Sep 14 2010

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

T. D. Noe and Robert Israel, Table of n, a(n) for n = 3..412 (3..100 from T. D. Noe)

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

Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012

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, 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.

a(n) = [(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

a(n) = det(S(i+3,j+2)|, 1 <= i,j <= n-3), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013

a(n) = Gamma(n)*(HarmonicNumber(n-1)^2+Zeta(2,n)-Zeta(2))/2. - Gerry Martens, Jul 05 2015

EXAMPLE

(-log(1-x))^3 = x^3 + 3/2*x^4 + 7/4*x^5 + 15/8*x^6 + ...

MAPLE

seq(abs(Stirling1(n, 3)), n=3..30); # Robert Israel, Jul 05 2015

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]

Abs[StirlingS1[Range[3, 30], 3]] (* Harvey P. Dale, Jun 23 2014 *)

f[n_] := Gamma[n]*(HarmonicNumber[n - 1]^2 + Zeta[2, n] - Zeta[2])/2; Array[f, 19, 3] (* Robert G. Wilson v, Jul 05 2015 *)

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, Jun 27 2008

(MAGMA) A000399:=func< n | Abs(StirlingFirst(n, 3)) >; [ A000399(n): n in [3..25] ]; // Klaus Brockhaus, Jan 14 2011

CROSSREFS

Cf. A000254, A000454, A000482, A001233, A001234, A243569, A243570, A008275.

Sequence in context: A144638 A117671 A213452 * A081051 A145145 A249476

Adjacent sequences:  A000396 A000397 A000398 * A000400 A000401 A000402

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 15:31 EDT 2017. Contains 284273 sequences.