This site is supported by donations to The OEIS Foundation.

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003011 Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times. (Formerly M3071) 5

%I M3071

%S 1,3,19,271,7365,326011,21295783,1924223799,229714292041,

%T 35007742568755,6630796801779771,1527863209528564063,

%U 420814980652048751629,136526522051229388285611

%N Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.

%C E.g.f. A(x)=y satisfies 0=(2x^3+2x^2)y''+(-3x^3+4x-1)y'+(x^3-x^2-2x+3)y. - _Michael Somos_, Mar 15 2004

%C Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a sequence of n (possibly empty) sets, each having at most 2 elements. - Bob Proctor, Apr 18 2005

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 17.

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

%H Robert A. Proctor, <a href="http://arxiv.org/abs/math.CO/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math.CO/0606404, Jan 05, 2007

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F a(n)n=a(n-1)(2n^3-n^2+n+1)+a(n-2)(-3n^3+4n^2+2n-3)+a(n-3)(n^3-2n^2-n+2).

%F a(n) ~ sqrt(Pi)*2^(n+1)*n^(2*n+1/2)/exp(2*n-1). - _Vaclav Kotesovec_, Oct 19 2013

%t Table[nn=2n;a=1+x+x^2/2!;Total[Range[0,nn]!CoefficientList[Series[a^n,{x,0,nn}],x]],{n,0,15}] (* _Geoffrey Critzer_, Dec 23 2011 *)

%o (PARI) a(n)=local(A);if(n<0,0,A=(1+x+x^2/2)^n;sum(k=0,2*n,k!*polcoeff(A,k)))

%Y a(n) = Sum[C(n, k)*A105749(k), 0<=k<=n]

%Y Replace "sequence" by "collection" in comment: A105748.

%Y Replace "sets" by "lists" in comment: A082765.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Aug 18 2002

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.