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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) 7
 1, 3, 19, 271, 7365, 326011, 21295783, 1924223799, 229714292041, 35007742568755, 6630796801779771, 1527863209528564063, 420814980652048751629, 136526522051229388285611 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 17. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 0..230 Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404, Jan 05, 2007 FORMULA 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). a(n) ~ sqrt(Pi)*2^(n+1)*n^(2*n+1/2)/exp(2*n-1). - Vaclav Kotesovec, Oct 19 2013 MATHEMATICA 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 *) PROG (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))) CROSSREFS a(n) = Sum[C(n, k)*A105749(k), 0<=k<=n] Replace "sequence" with "collection" in comment: A105748. Replace "sets" with "lists" in comment: A082765. Sequence in context: A316294 A233240 A173799 * A231620 A268646 A143597 Adjacent sequences:  A003008 A003009 A003010 * A003012 A003013 A003014 KEYWORD nonn AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Aug 18 2002 STATUS approved

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Last modified May 26 13:49 EDT 2020. Contains 334626 sequences. (Running on oeis4.)