5,2

Number of permutations of n elements with exactly 5 cycles.

Let P(n+3,X)=(X+1)(X+2)(X+3)...(X+n+3); then a(n) is the coefficient of X^4; or a(n)=P''''(n+3,0)/4! - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2002

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.

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=5..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].

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=5,n=1) ~ exp(-x)/x^5*(1 - 15/x + 175/x^2 - 1960/x^3 + 22449/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion.

(End)

E.g.f. (-log(1-x))^5/5! or (1-x)^-1 * (-log(1-x))^4 [Corrected by Joerg Arndt, Oct 05 2009]

a(n) is coefficient of x^(n+5) in (-log(1-x))^5, multiplied by (n+5)!/5!

(-log(1-x))^5 = x^5 + 5/2*x^6 + 25/6*x^7 + 35/6*x^8 + ...

(PARI) for(n=4, 50, print1(polcoeff(prod(i=1, n, x+i), 4, x), ", "))

sage: [stirling_number1(i, 5) for i in xrange(5, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A000254, A000399, A000454, A001233, A008275 (Stirling1 triangle).

Sequence in context: A036083 A051588 A016164 this_sequence A145147 A069379 A120995

Adjacent sequences: A000479 A000480 A000481 this_sequence A000483 A000484 A000485

nonn,easy,more

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