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

1, 3, 19, 271, 7365, 326011, 21295783, 1924223799, 229714292041, 35007742568755, 6630796801779771, 1527863209528564063, 420814980652048751629, 136526522051229388285611

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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
• Index entries for related partition-counting sequences

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.

Programs

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 *)

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)))

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

Extensions

More terms from Vladeta Jovovic, Aug 18 2002