A000276 - OEIS

%I M3075 N1248 #52 Sep 25 2016 04:37:26

%S 3,20,130,924,7308,64224,623376,6636960,76998240,967524480,

%T 13096736640,190060335360,2944310342400,48503818137600,

%U 846795372595200,15618926924697600,303517672703078400,6198400928176128000,132720966600284160000,2973385109386137600000

%N Associated Stirling numbers.

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

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

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Shanzhen Gao, Permutations with Restricted Structure (in preparation).

%H Alois P. Heinz, <a href="/A000276/b000276.txt">Table of n, a(n) for n = 4..150</a>

%F a(n) = (n-1)!*Sum_{i=2..n-2} 1/i = (n-1)!*(Psi(n-1)+gamma-1). - _Vladeta Jovovic_, Aug 19 2003

%F With alternating signs: Ramanujan polynomials psi_3(n-2, x) evaluated at 1. - _Ralf Stephan_, Apr 16 2004

%F E.g.f.: ((x+log(1-x))^2)/2. [Corrected by _Vladeta Jovovic_, May 03 2008]

%F 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

%F 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

%F 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

%F 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

%F a(n) = A000254(n-1) - (n-1)! - (n-2)!. - _Anton Zakharov_, Sep 24 2016

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

%t 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 *)

%t a[n_] := (n-1)!*(HarmonicNumber[n-2]-1); Table[a[n], {n, 4, 23}] (* _Jean-François Alcover_, Feb 06 2016, after _Gary Detlefs_ *)

%o (PARI) a(n) = (n-1)!*sum(i=2, n-2, 1/i); \\ _Michel Marcus_, Feb 06 2016

%Y A diagonal of triangle in A008306.

%Y Cf. A052518, A052881, A259456.

%K nonn

%O 4,1

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_