4,2

Number of permutations of n elements with exactly 4 cycles.

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=4..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 33

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

E.g.f.: (-log(1-x))^4/4! or (1-x)^-1 * (-log(1-x))^3. [Corrected by Joerg Arndt, Oct 05 2009]

a(n) is coefficient of x^(n+4) in (-log(1-x))^4, multiplied by (n+4)!/4!

[h(n-1, 1)^3-3*h(n-1, 1)*h(n-1, 2)+2*h(n-1, 3)]*(n-1)!/3!, h(n, r)=sum(1/i^r, i=1..n).

(-log(1-x))^4 = x^4 + 2*x^5 + 17/6*x^6 + 7/2*x^7 + ...

(PARI) for(n=3, 50, print1(polcoeff(prod(i=1, n, x+i), 3, x), ", "))

sage: [stirling_number1(i, 4) for i in xrange(4, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

Cf. A000254, A000399, A000482, A001233, A008275 (Stirling1 triangle).

Sequence in context: A081903 A144639 A038235 this_sequence A145146 A163412 A121115

Adjacent sequences: A000451 A000452 A000453 this_sequence A000455 A000456 A000457

nonn

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

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 18 2000