A000276 Associated Stirling numbers. (Formerly M3075 N1248)

3, 20, 130, 924, 7308, 64224, 623376, 6636960, 76998240, 967524480, 13096736640, 190060335360, 2944310342400, 48503818137600, 846795372595200, 15618926924697600, 303517672703078400, 6198400928176128000, 132720966600284160000, 2973385109386137600000

Offset

4,1

Comments

a(n) is also the number of permutations of n elements, without any fixed point, with exactly two cycles. - Shanzhen Gao, Sep 15 2010

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 4..150

Formula

a(n) = (n-1)!*Sum_{i=2..n-2} 1/i = (n-1)!*(Psi(n-1)+gamma-1). - Vladeta Jovovic, Aug 19 2003

With alternating signs: Ramanujan polynomials psi_3(n-2, x) evaluated at 1. - Ralf Stephan, Apr 16 2004

E.g.f.: ((x+log(1-x))^2)/2. [Corrected by Vladeta Jovovic, May 03 2008]

a(n) = Sum_{i=2..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 15 2010

a(n) = (n+3)!*(h(n+2)-1), with offset 0, where h(n)=sum(1/k,k=1..n). - Gary Detlefs, Sep 11 2010

Conjecture: (-n+2)*a(n) +(n-1)*(2*n-5)*a(n-1) -(n-1)*(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 18 2015

Conjecture: a(n) +2*(-n+2)*a(n-1) +(n^2-6*n+10)*a(n-2) +(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 18 2015

a(n) = A000254(n-1) - (n-1)! - (n-2)!. - Anton Zakharov, Sep 24 2016

Example

a(4) = 3 because we have: (12)(34),(13)(24),(14)(23). - Geoffrey Critzer, Nov 03 2012

Mathematica

nn=25; a=Log[1/(1-x)]-x; Drop[Range[0, nn]!CoefficientList[Series[a^2/2, {x, 0, nn}], x], 4]  (* Geoffrey Critzer, Nov 03 2012 *)
a[n_] := (n-1)!*(HarmonicNumber[n-2]-1); Table[a[n], {n, 4, 23}] (* Jean-François Alcover, Feb 06 2016, after Gary Detlefs *)

Prog

(PARI) a(n) = (n-1)!*sum(i=2, n-2, 1/i); \\ Michel Marcus, Feb 06 2016

Crossrefs

A diagonal of triangle in A008306.

Cf. A052518, A052881, A259456.

Sequence in context: A167590 A228884 A138910 * A216778 A337105 A056306

Adjacent sequences: A000273 A000274 A000275 * A000277 A000278 A000279

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower

Status

approved