A000276 Associated Stirling numbers.

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

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

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

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

Crossrefs

A diagonal of triangle in A008306.

Cf. A052518, A052881, A259456.

Programs

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 *)

PARI

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

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

Extensions

More terms from Christian G. Bower