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A003011 Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.
(Formerly M3071)

%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 G. C. Greubel, <a href="/A003011/b003011.txt">Table of n, a(n) for n = 0..230</a>

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

%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

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Last modified July 27 22:07 EDT 2017. Contains 289866 sequences.