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

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

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

Index entries for related partition-counting sequences

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 * A338772 A231620 A268646

Adjacent sequences: A003008 A003009 A003010 * A003012 A003013 A003014

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Aug 18 2002

Status

approved