3,2

Number of permutations of n elements with exactly 3 cycles.

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=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.

(End)

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).

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

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! or (1-x)^(-1) * (-log(1-x))^2. [Corrected by Joerg Arndt, Oct 05 2009]

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

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

(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

Cf. A000254, A000454, A000482, A001233, A008275 (Stirling1 triangle).

Sequence in context: A001109 A144638 A117671 this_sequence A081051 A145145 A087631

Adjacent sequences: A000396 A000397 A000398 this_sequence A000400 A000401 A000402

nonn,easy,nice

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